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Scientific Notation - Practice Problems

Solving earth science problems with scientific notation, × div[id^='image-'] {position:static}div[id^='image-'] div.hover{position:static} introductory problems.

These problems cover the fundamentals of writing scientific notation and using it to understand relative size of values and scientific prefixes.

Problem 1: The distance to the moon is 238,900 miles. Write this value in scientific notation.

Problem 2: One mile is 1609.34 meters. What is the distance to the moon in meters using scientific notation?

`1609.34 m/(mi) xx 238","900 mi` = 384,400,000 m

Notice in the above unit conversion the 'mi' units cancel each other out because 'mi' is in the denominator for the first term and the numerator for the second term

Problem 4: The atomic radius of a magnesium atom is approximately 1.6 angstroms, which is equal to 1.6 x 10 -10 meters (m). How do you write this length in standard form?

0.00000000016 m

Fissure A = 40,0000 m Fissure B = 5,0000 m

This shows fissure A is larger (by almost 10 times!). The shortcut to answer a question like this is to look at the exponent. If both coefficients are between 1-10, then the value with the larger exponent is the larger number.

Problem 6: The amount of carbon in the atmosphere is 750 petagrams (pg). One petagram equals 1 x 10 15 grams (g). Write out the amount of carbon in the atmosphere in (i) scientific notation and (ii) standard decimal format.

The exponent is a positive number, so the decimal will move to the right in the next step.

750,000,000,000,000,000 g

Advanced Problems

Scientific notation is used in solving these earth and space science problems and they are provided to you as an example. Be forewarned that these problems move beyond this module and require some facility with unit conversions, rearranging equations, and algebraic rules for multiplying and dividing exponents. If you can solve these, you've mastered scientific notation!

Problem 7: Calculate the volume of water (in cubic meters and in liters) falling on a 10,000 km 2 watershed from 5 cm of rainfall.

`10,000 km^2 = 1 xx 10^4 km^2`

5 cm of rainfall = `5 xx 10^0 cm`

Let's start with meters as the common unit and convert to liters later. There are 1 x 10 3 m in a km and area is km x km (km 2 ), therefore you need to convert from km to m twice:

`1 xx 10^3 m/(km) * 1 xx 10^3 m/(km) = 1 xx 10^6 m^2/(km)^2` `1 xx 10 m^2/(km)^2 * 1 xx 10^4 km^2 = 1 xx 10^10 m^2` for the area of the watershed.

For the amount of rainfall, you should convert from centimeters to meters:

`5 cm * (1 m)/(100 cm)= 5 xx 10^-2 m`

`V = A * d`

When multiplying terms with exponents, you can multiply the coefficients and add the exponents:

`V = 1 xx 10^10 m^2 * 5.08 xx 10^(-2) m = 5.08 xx 10^8 m^3`

Given that there are 1 x 10 3 liters in a cubic meter we can make the following conversion:

`1 xx 10^3 L * 5.08 xx 10^8 m^3 = 5.08 xx 10^11 L`

Step 5. Check your units and your answer - do they make sense?

`V = 4/3 pi r^3`

Using this equation, plug in the radius (r) of the dust grains.

`V = 4/3 pi (2 xx10^(-6))^3m^3`

Notice the (-6) exponent is cubed. When you take an exponent to an exponent, you need to multiply the two terms

`V = 4/3 pi (8 xx10^(-18)m^3)`

Then, multiple the cubed radius times pi and 4/3

`V = 3.35 xx 10^(-17) m^3`

`m = 3.35 xx 10^(-17) m^3 * 3300 (kg)/m^3`

Notice in the equation above that the m 3 terms cancel each other out and you are left with kg

`m = 1.1 xx 10^(-13) kg`

`V = 4/3 pi (2.325 xx10^(15) m)^3`

`V = 5.26 xx10^(46) m^3`

Number of dust grains = `5.26 xx10^(46) m^3 xx 0.001` grains/m 3

Number of dust grains = `5.26xx10^43 "grains"`

Total mass = `1.1xx10^(-13) (kg)/("grains") * 5.26xx10^43 "grains"`

Notice in the equation above the 'grains' terms cancel each other out and you are left with kg

Total mass = `5.79xx10^30 kg`

If you feel comfortable with this topic, you can go on to the assessment . Or you can go back to the Scientific Notation explanation page .

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How to Solve Scientific Notation? (+FREE Worksheet!)

Learn how to solve mathematics problems containing scientific notation in a few simple and easy steps.

Step by step guide to solve scientific notation problems

Scientific notation is used to write very big or very small numbers in decimal form.
In scientific notation all numbers are written in the form of: $m×10^n$

Scientific Notation – Example 1:

Write $0.00015$ in scientific notation.

First, move the decimal point to the right so that you have a number that is between $1$ and $10$. Then: $N=1.5$ Second, determine how many places the decimal moved in step $1$ by the power of $10$. Then: $10^{ \ -4} →$ When the decimal is moved to the right, the exponent is negative. Then: $0.00015=1.5×10^{ \ -4} $

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Scientific notation – example 2:.

Write $9.5 \times 10^{\ -5}$ in standard notation.

$10^{-5} →$ When the decimal is moved to the right, the exponent is negative. Then: $9.5×10^{-5}=0.000095$

Scientific Notation – Example 3:

Write $0.00012$ in scientific notation.

First, move the decimal point to the right so that you have a number that is between $1$ and $10$. Then: $N=1.2$ Second, determine how many places the decimal moved in step $1$ by the power of $10$. Then: $10^{-4}→ $ When the decimal is moved to the right, the exponent is negative. Then: $0.00012=1.2×10^{-4}$

Scientific Notation – Example 4:

Write $8.3×10^{-5}$ in standard notation.

$10^{-5} →$ When the decimal is moved to the right, the exponent is negative. Then: $8.3×10^{-5}=0.000083$

Exercises for Solving Scientific Notation

Write each number in scientific notation..

$\color{blue}{91 × 10^3}$
$\color{blue}{60}$
$\color{blue}{2000000}$
$\color{blue}{0.0000006}$
$\color{blue}{354000}$
$\color{blue}{0.000325}$

Download Scientific Notation Worksheet

$\color{blue}{9.1 × 10^4}$
$\color{blue}{6 × 10^1}$
$\color{blue}{2 × 10^6}$
$\color{blue}{6 × 10^{–7}}$
$\color{blue}{3.54 × 10^5}$
$\color{blue}{3.25 × 10^{–4}}$

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A Complete Guide to Scientific Notation (Standard Form)

Scientific notation: video lesson, what is scientific notation.

Scientific notation can be used to represent both decimals less than one whole and very large whole numbers.

Scientific notation is a concise way to represent both very large or very small numbers. Scientific notation involves a number between 1 and 10 being multiplied to a power of 10. The power of 10 is positive if the number being represented is large and it is negative if the number is small.

A positive power of 10 tells us how many digits the number has between its first digit and the decimal point
A negative power of 10 tells us how many zero digits are in front of the non-zero digits.

For example, 6×10 5 is a 6 followed by 5 digits. It represents 600 000.

For example, 6×10 -5 is a 6 with 5 zeros before it. It represents 0.00006.

How Scientific Notation is Used in Real Life

Scientific notation is used whenever very large or very small numbers are involved. For example in physics, astronomy, chemistry and finance.

Some examples of scientific notation used in real-life include:

The very large distances involved in astronomy. For example, the distance between the Earth and the Sun is said to be 93 million miles, which can be written as 9.3 × 10 7 miles.
In Physics, very large or very small known constants are represented in scientific notation such as the speed of light (2.998 × 10 8 meters per second) or Planck’s constant (6.63 × 10 -34 joule-seconds).
In chemistry, the mass of atoms, molecules and molar mass is often written in scientific notation. For example, Avogadro’s number is 6.02 × 10 23 particles per mole.
In finance and economics, figures such as national debt, gross domestic product (GDP), or market capitalisation are often expressed in scientific notation. For instance, if a country’s GDP is $2.5 trillion, it would be expressed as 2.5 × 10 12 dollars.
Engineers often work with values that span many orders of magnitude, such as voltage, current, or power.
In medicine and biology, scientific notation is used to express values like cell counts, DNA base pairs, or concentrations of substances in body fluids.

In science, abbreviations are commonly used to refer to given orders of magnitude such as:

Here are some common values of scientific notation listed on a number line from least to greatest.

Scientific Notation Explained

Scientific notation is used when a very large or very small number is too long to write in the normal manner. In scientific notation numbers are written as a number between 1 and 10 multiplied by a power of 10. For example, 32500 is written as 3.25×10 4 .

Scientific notation is a more concise way to write long numbers whilst indicating their order of magnitude.

Scientific notation is also commonly known as standard form. Both scientific notation and standard form refer to numbers written as a number between 1 and 10 multiplied by 10 to the power of some other positive or negative number.

For example consider the following lists:

10 3 = 1000
10 4 = 10000

The power that 10 is raised to in each case is equal to the number of digits after the 1.

3 × 10 1 = 30
3 × 10 2 = 300
3 × 10 3 = 3000
3 × 10 4 = 30000

The power that 10 is raised to in each case is equal to the number of digits after the 3.

A number written in scientific notation always consists of the following parts:

A number between the values of 1 and 10
A multiplication by 10 raised to a power

The name for a number not written in scientific notation is simply ‘standard notation’ or ‘decimal notation’. When comparing a number written in standard notation to a number written in scientific notation, it may also be referred to as ‘expanded form’.

The first step is to write the first part of the number which must always be between 1 and 10.

We take 32500 and write it as 3.25 so that it is now a number that is bigger than 1 and less than 10.

The next step is to find the power of 10 that 3.25 is raised to to make it equal to 32500.

The power that ten is raised to is equal to the number of digits after the first digit.

In 32500, there are 4 digits after the first digit of 3. Therefore the power that 10 is raised to is 4.

32500 can be written as 3.25 × 10 4 .

Scientific notation is often used on calculator displays (since the calculator screen cannot fit very long numbers). On a calculator, scientifi notation is written using ‘E’ to represent × 10^.

For example, 32500 can be written as 3.25 × 10 4 or alternatively as 3.25E4.

In scientific notation, E stands for × 10 raised to the power of the number that comes after the E.

Scientific notation can also be used to write very small numbers by using negative powers of 10.

We can see that the negative power of 10 describes how many zeros are at the start of the decimal number.

For example, 10 -3 has 3 zeros when written as 0.001.

scientific notation with negative exponents

The number 0.000034 can be written as 3.4 × 10 -5 . There are 5 zeros before the digits of 3 and 4.

how to write a very small decimal number in scientific notation

Rules of Scientific Notation

Numbers written in scientific notation must adhere to the following rules:

The number before the multiplication sign must always be at least 1 and less than 10.
There is always a multiplication by a power of 10 (as opposed to a multiplication by any other number).
The power of 10 can be negative, zero or positive.
If the number being represented is 10 or more, the power of 10 is positive.
If the number being represented is 10 or more, the power of 10 is equal to the number of digits that come after the first digit of the number.
If the number being represented is less than 1, the power of 10 is negative.
If the number being represented is less than 1, the power of 10 is equal to (-1) multiplied by the number of zeros at the start of the number.
If the number being represented is greater than 1 but less than 10, the power of 10 is zero.

How to Write Large Numbers in Scientific Notation

Write a decimal point after the first digit of the number to form a number between 1 and 10.
Multiply this by a power of 10, where the power is equal to the number of digits after the first digit of the large number.

For example, write 630000 in scientific notation:

Step 1. Write a decimal point after the first digit of the number to form a number between 1 and 10

A decimal point is placed after the 6 to make 6.30000 which is just 6.3

6.3 is a number between 1 and 10.

Step 2. Multiply this by a power of 10, where the power is equal to the number of digits after the first digit of the large number

The first digit of 630000 is 6. We count the number of digits after this first digit of 6.

There are 5 digits after the first digit. That is, the 3, 0, 0, 0 and 0.

Therefore we multiply by 10 5 .

630000 is written in scientific notation as 6.3×10 5 .

how to write a number in scientific form

How to Write Small Numbers in Scientific Notation

Write the non-zero digits as a number between 1 and 10.
Multiply this by a negative power of 10 equal to the number of zeros in front of the first non-zero digit.

For example, write 0.000008 in scientific notation.

Step 1. Write the non-zero digits as a number between 1 and 10

The zeros at the start of 0.000008 are ignored to obtain the non-zero digit of 8.

8 is written because it is a number between 1 and 10.

Step 2. Multiply this by a negative power of 10 equal to the number of zeros in front of the first non-zero digit

The first non-zero digit in 0.000008 is the 8.

The number of zeros in front of this non-zero digit is 6.

Since there are 6 zeros, we multiply the 8 by 10 to the power of -6.

0.000008 is written in scientific notation as 8×10 -6 .

how to write a small number in standard form

Small numbers less than one whole are written in scientific notation with a negative exponent representing the number of zeros at the start of the number.

For example, write the number 0.0257 in scientific notation.

The zeros at the start of 0.0257 are ignored and the decimal point is placed after the first non-zero digit of 2.

This results in 2.57 which is a number between 1 and 10.

The first non-zero digit is 2.

There are 2 zeros before the 2 and so, we multiply 2.57 by 10 to the power of -2.

0.0257 is written in scientific notation as 2.57×10 -2 .

how to write a very small numbers in scientific notation

The power of the 10 is equal to the number of zeros at the start of the number multiplied by -1.

How to Read Scientific Notation

If the power of 10 is positive, this is the number of digits after the first digit. So 2.11×10 6 has 6 digits after the 2. It is written as 2110000.
If the power of 10 is negative, this is the number of zeros in front of the digits. So 5.3 × 10 -3 has 3 zeros at the start. It is written as 0.0053.

Scientific Notation for a Large Number

The power of 10 is 6 and so, there are 6 digits after the 2.

We already have two digits of 1 and so, four more 0 digits are needed.

Scientific Notation for a Small Number

how to read scientific notation with negative exponent

The power of 10 is negative 3.

Negative powers tell us how many 0 digits are at the start of the number.

We put three 0 digits and then the 5 and the 3.

The decimal point always comes after the first 0 digit.

For example, 2.93 × 10 4 has 4 digits after the first digit of 2.

Following the 2, there is a 9 then a 3. We need two more 0 digits to obtain 4 digits after the 2.

2.93 × 10 4. = 29300.

For example, 1.04 × 10 -4 has a negative power of 10.

Therefore this power is equivalent to the number of zeros at the start of the number. There will be 4 zeros followed by the digits of 1, 0 and 4.

1.04 × 10 -4 = 0.000104.

Notice that the power of 10 is only equal to the number of zeros at the start of the number, not the total number of zeros in the answer.

standard form with negative powers of 10

Examples of Scientific Notation

Notice that as an extra zero digit is added to the value, the exponent in scientific notation increases by 1.

Notice that as we move from one thousand to one million to one billion to one trillion, three zero digits are added and so, the exponent in scientific notation increases by 3.

Here are some common values listed in scientific notation:

The power of zero in 1×10 0 means that zero tens have been multiplied by.

That is, 10 0 = 1 and so 1×10 0 simply means 1×1 which is just equal to 1.

Here are some examples of numbers written in scientific notation:

How to Round Numbers in Scientific Notation

It is common for numbers to be written to a particular number of significant figures and then written in scientific notation. This is because numbers in scientific notation are very large or very small and so, the final digits do not really impact on the overall size of the number.

Typically, numbers in scientific notation are given to 2 or 3 decimal places although further accuracy may be required.

It is common to round numbers in scientific notation to 3 significant figures.

For example, write 125364 in scientific notation, rounded to 3 significant figures.

The first step is to round the number to 3 significant figures.

That is, we look at the first 3 non-zero digits of the number, which are 1, 2 and 5. We then look at the next digit after this to decide if the 3rd significant figure of 5 remains as a 5 or rounds up to a 6.

Only round up the 3rd significant figure if the 4th significant figure is equal to 5 or more.

Since the digit after the 5 is a 3, we do not round up.

We write 125 and then replace the other digits with zeros, so that 125364 written to 3 significant figures is 125000.

We now write this in scientific notation as 1.25 × 10 5 , since there are 5 digits after the first digit of 1 in the number 125000.

how to round a number to 3 significant figures in scientific notation

For example, write the number 0.06627 in scientific notation to 3 significant figures.

Significant figures are counted after the zeros at the start of the number.

We look at the first 3 significant figures of the number which are 6, 6 and 2.

We only round up the 3rd significant figure if the 4th significant figure is equal to 5 or more.

The 4th significant figure is a 7 and so, we round the 2 up to a 3.

0.06627 rounded to 3 significant figures is 0.0663.

This number starts with 2 zeros, so writing it in scientific notation we have 6.33 × 10 -2 .

how to write a decimal number in scientific notation to 3 significant figures

How to Add and Subtract Numbers in Scientific Notation

The ease of adding and subtracting numbers in scientific notation depends on whether the size of the exponents are the same in each number.

To add or subtract numbers in scientific notation that have the same exponent, simply add or subtract the coefficients and keep the exponent the same. For example, 3×10 5 + 4 × 10 5 is (3+4) × 10 5 which equals 7 × 10 5 .

In this example, the exponents on both 3×10 5 and 4×10 5 are both 5.

Therefore, we can simply add the 3 and the 4 together to obtain 7, whilst the exponent in the answer remains as 5.

3×10 5 + 4×10 5 = 7×10 5 .

This addition is essentially the same as 300 000 + 400 000 = 700 000.

how to add numbers in scientific notation with the same exponent

For example, 3×10 -4 – 1×10 -4 = (3-1)×10 -4 which is just 2×10 -4 .

In this example, we simply subtract the coefficients and keep the exponent as -4.

To add or subtract numbers in scientific notation that have different exponents, it is easiest to convert the numbers to standard notation first. Then perform the addition or subtraction and write the result back in scientific notation if needed.

In the example of 3.21×10 3 + 2×10 4 , the exponents are different sizes.

Converting 3.21×10 3 to standard notation, we have 3 210. The exponent of 3 means there are 3 digits after the first digit.

After the 3, we have the 2 then the 1 and so, another digit of 0 is needed to make 3 digits after the first digit in total.

Converting 2×10 4 to standard notation, we have 20 000. That is, we have 4 digits after the first digit of 2.

Now, these numbers can be added following the standard addition process.

3 210 + 20 000 = 23 210.

We can write this in scientific notation if needed as 2.321×10 4 since there are 4 digits after the first digit in 23 210.

how to add numbers in scientific notation

Here is an alternate method for adding numbers in scientific notation when they have different exponents.

The same numbers as above are used but here we are demonstrating a different method to solve the problem.

To add or subtract numbers in scientific notation, first convert the numbers to have the same exponent. To do this, multiply the coefficient of the number with the largest exponent by 10 each time the exponent is reduced by 1.

For example, in 3.21×10 3 + 2×10 4 , we wish to reduce the exponent in 2×10 4 from a 4 to a 3 so that it is the same exponent as in 3.21×10 3 .

Since we need to reduce the exponent by 1, we multiply the 2 by 10 to make 20.

2×10 4 is the same as 20×10 3 .

Therefore 3.21×10 3 + 2×10 4 is rewritten as 3.21×10 3 + 20×10 3 .

Now the coefficients can be added so that 3.21×10 3 + 20×10 3 = (3.21+20)×10 3 .

This equals 23.21×10 3 which can be readjusted to 2.321×10 4 as we divide the coefficient by 10 as we increase the exponent of 10 by 1.

Here is an example of subtracting numbers in scientific notation by first converting them to have the same exponents.

In the example of 5.3×10 5 – 7.9×10 4 , we need to change the exponent of 5.3×10 5 to a 4.

To do this, we multiply the 5.3 by 10 to make 53.

5.3×10 5 is the same as 53×10 4 .

Therefore 5.3×10 5 – 7.9×10 4 can be written as 53×10 4 – 7.9×10 4 .

Now the numbers have the same exponent, subtract the coefficients like so:

(53 – 7.9)×10 4 = 45.1×10 4 .

Finally, this is rewritten so that the coefficient is a number between 1 and 10. We divide 45.1 by 10 and increase the exponent by 1 to compensate.

45.1×10 4 = 4.51×10 5 .

how to subtract numbers in standard form

How to Multiply and Divide Numbers in Scientific Notation

To multiply numbers in scientific notation, multiply the coefficients and add the exponents. For example, (2×10 3 ) × (3×10 5 ) = 6×10 8 . The coefficients of 2 and 3 were multiplied to make 6 and the exponents of 3 and 5 were added to make 8.

This occurs since exponents are always added when they are multiplied together.

how to multiply numbers in scientific notation

Here is an example of multiplying numbers in scientific notation: (4.5×10 9 ) × (5.2×10 -2 ).

Firstly, the coefficients of 4.5 and 5.2 are multiplied to obtain 23.4.

Secondly, the exponents are added so that 9 + -2 = 7.

Therefore, (4.5×10 9 ) × (5.2×10 -2 ) = 23.4×10 7 .

However, 23.4 must be written as a number between 1 and 10.

We divide 23.4 by 10 to obtain 2.34 and we increase the exponent from 7 to 8 to compensate.

(4.5×10 9 ) × (5.2×10 -2 ) = 2.34×10 8 .

how to multiply numbers written in scientific notation

To divide numbers in scientific notation, divide the coefficients and subtract the exponents. For example, (6×10 7 ) ÷ (2×10 2 ) = 3×10 5 . The coefficients of 6 and 2 were divided to obtain 3 and the exponent of 2 was subtracted from the exponent of 7 to obtain 5.

This occurs since exponents are subtracted when they a division takes place.

how to divide numbers in scientific notation

Here is another example of dividing numbers in scientific notation: (8.4×10 4 ) ÷ (2.5×10 -3 ).

Firstly, the coefficients are divided so 8.4 ÷ 2.5 = 3.36.

Secondly, the exponents are subtracted. 4 – -3 is the same as 4 + 3 which equals 7.

(8.4×10 4 ) ÷ (2.5×10 -3 ) = 3.36×10 7 .

Scientific Notation on Calculator

Scientific notation is commonly used in calculator displays.

To write scientific notation on a calculator, use the button labelled as ‘×10 𝑥 ‘ or just 10 𝑥 ‘.

Enter the coefficient (the number between 1 and 10)
Then press the ‘×10 𝑥 ‘ button
Then enter the power of 10

Here are the instructions for entering scientific notation on some common calculators:

Casio Fx-300

The ‘×10 𝑥 ‘ button is found in the middle of the bottom row.

Casio Fx-82

Casio fx-cg50.

The ’10 𝑥 ‘ option is found above the ‘log’ button. Press ‘2nd’ then ‘log’ to write numbers in scientific notation.

Press the ‘2nd’ button
Then press the ‘log’ button to select the ’10 𝑥 ‘ option

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Scientific notation

Here you will learn about scientific notation including how to convert between ordinary numbers and scientific notation and how to calculate with numbers in scientific notation.

Students will first learn about scientific notation as part of expressions and equations in 8 th grade.

What is scientific notation?

Scientific notation is a way of writing very large or very small numbers by using powers of ten.

Numbers in scientific notation are written in this format:

a\times10^{n}

Where a is a number 1\leq{a}<10 and n is an integer (whole number).

To write a number in scientific notation, you need to understand the place value of the number.

For example, let’s look at the number 8,290,000 and write the digits in a place value table:

So 8,290,000 written in scientific notation is 8.29\times10^{6} .

Step-by-step guide: Powers of 10

Scientific notation is a representation of place value which compliments the decimal number system, as shown in the table below.

Any integer or terminating decimal can be written using the scientific notation a\times10^{n}. The table below shows how the value of a can remain the same while the power of ten ‘n’ changes the place value of those digits.

Using scientific notation enables us to write very large or very small numbers.

For example,

Using scientific notation also enables us to compare the size of very large or very small numbers easily.

Which is larger: 8,560,000,000,000 or 45,320,000,000,000?

At a glance, it is difficult to tell which is larger, but written in scientific notation, you can compare these numbers very quickly, as shown.

Instantly you can see that 4.532\times10^{13} is the larger number as it has the higher power of ten.

Scientific notation calculation

When numbers are written in scientific notation it can make some calculations neater and quicker to compute.

For example, solve 8.56\times{10^5}-2.3\times{10^2}.

How do you begin? If you expand each number, the expression is 856,000-230.

Notice that none of the digits are in the same place value. So, to solve using scientific notation, convert so that all numbers have the same power of 10.

Now the expression is 8,560\times{10^2}-2.3\times{10^2}.

Finally, convert 8,557.7\times{10^2} back to scientific notation.

Step-by-step guide: Adding and subtracting scientific notation

Now let’s solve 3.4\times{10^{5}}\times{2}\times{10^{-3}}.

Since multiplication is commutative (the order does not matter), you can reorder this calculation to 3.4\times{2}\times{10^{5}}\times{10^{-3}}.

Hint: Use the rule a^{b}\times{a^{c}}=a^{b+c} to simplify the powers of 10.

Since 6.8 is between 1 and 10, you don’t need to adjust the power of 10.

Step-by-step guide: How to multiply scientific notation

Now let’s solve \left(3.4\times{10^5}\right)\div\left(2\times{10^3}\right).

Re–write this expression as \cfrac{3.4\times{10^5}}{2\times{10^3}}, which equals \cfrac{3.4\times{10}\times{10}\times{10}\times{10}\times{10}}{2\times{10}\times{10}\times{10}}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

Since 1.7 is between 1 and 10, you don’t need to adjust the power of 10.

Step-by-step guide: How to divide scientific notation

Common Core State Standards

How does this relate to 8 th grade math?

Grade 8 – Expressions and Equations (8.EE.A.3) Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3\times{10^8} and the population of the world as 7\times{10^9}, and determine that the world population is more than 20 times larger.
Grade 8 – Expressions and Equations (8.EE.A.4) Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

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How to write numbers with scientific notation

In order to represent a number in scientific notation:

Show the number as an expression with multipliers of \bf{10}.
Determine the power of \bf{10}.
Write the number.

Scientific notation examples

Example 1: writing numbers in scientific notation.

Write this number in scientific notation: 52,000

Remember the first number is always between 1 and 10 .

2 Determine the power of \bf{10}.

3 Write the equation.

How to convert numbers written in scientific notation to ordinary numbers

In order to solve problems with numbers written in scientific notation:

Use place value reasoning to identify how the power of \bf{10} will change the number.

Shift the digits left if multiplying or right if dividing.

Example 2: writing a number in scientific notation as an ordinary number

Write 9.4\times10^{5} as an ordinary number.

Each digit must move 5 places to the left.

How to add and subtract in scientific notation

In order to add and subtract numbers in scientific notation:

Convert the number(s) to have the same power of \bf{10}.

Add or subtract the non-zero digits.

Check your answer is in scientific notation.

Example 3: adding numbers in scientific notation

Calculate 6\times10^{4}+2\times10^{3}. Write your answer in scientific notation.

Let’s convert 2\times10^{3} to be 10^4.

To do this, multiply it by 10 to add one to the power. To maintain the value of the number you need to divide the non-zero number by 10.

Since 6.2 is between 1 and 10, you don’t need to adjust the power of 10.

Example 4: subtracting numbers in scientific notation

Calculate 5\times{10^8}-4\times{10^6}. Write your answer in scientific notation.

Let’s convert 5\times{10^8} to be 10^6.

5\times{10^8}=50\times{10^7}=500\times{10^6}

496 is not between 1 and 10.

Convert 496\times{10^6} back to scientific notation.

How to multiply and divide in scientific notation

In order to multiply and divide in scientific notation:

Multiply or divide the non-zero numbers.

Multiply or divide the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

Example 5: multiplying numbers in scientific notation

Calculate \left(5\times{10^4}\right)\times\left(7\times{10^8}\right).

Write your answer in scientific notation.

35 is not between 1 and 10.

Convert 35\times{10^{12}} back to scientific notation.

\left(5\times{10^4}\right)\times\left(7\times{10^8}\right)=3.5\times{10^{13}}

Example 6: dividing numbers in scientific notation

Calculate 8\times{10^7}\div{2}\times{10^2}. Write your answer in scientific notation.

Teaching tips for scientific notation

Give students real world examples of when scientific notation is useful, such as the distance from the sun to Neptune or the size of a microorganism.
Start with smaller numbers when students are first learning. This allows students an opportunity to make sense of problems or check their work by expanding numbers written in scientific notation, until they understand the rules for operating.
Allow struggling students to use a scientific notation calculator.

Easy mistakes to make

Not converting numbers to the same power of \bf{10} when adding or subtracting Multiplying by different powers of 10 means the first part of the number does not represent the same place value. In order to use the subtraction algorithm, the first number needs to represent the same place value positions. For example, 5.6\times{10^3}+6.7\times{10^2} 5.6\times{10^3} is read as “ 5.6 thousands” and 6.7\times{10^2} is read as “ 6.7 hundreds.” In order to add them, they need to represent the same place value. Since, 5.6\times{10^3}=56\times{10^2}, you can solve 56\times{10^2}+6.7\times{10^2}
Not converting solutions to scientific notation After calculating with scientific notation, a common mistake is to not check that the first part of the number is 1\leq{n}<10.
Forgetting the meaning of negative exponents Remember that decimals are represented as smaller powers of 10. The negative exponent represents the equivalent fractional value and is used for every position to the right of the decimal. For example, 10^{-1}=\cfrac{1}{10} 10^{-2}=\cfrac{1}{100} 10^{-3}=\cfrac{1}{1,000} 10^{-4}=\cfrac{1}{10,000}

Practice scientific notation questions

1. Write 86,000 in scientific notation.

The number between 1 and 10 here is 8.6. Since 8 is in the ten thousands column,

2. Write 0.0097 in scientific notation.

The number between 1 and 10 here is 9.7. Since 9 is in the thousandths position

3. Write 5.9\times{10^3} as an ordinary number.

10^{3}=1,000 therefore

4. Solve 7\times{10^5}+2\times{10^4}. Write your answer in scientific notation.

Before adding, the first number needs to have the same place values. To do this, both powers of 10 need to be the same.

Since 2\times{10^4}=0.2\times{10^5}, solve 7\times{10^5}+0.2\times{10^5}.

5. Solve \left(6\times{10^8}\right)\times\left(3\times{10^4}\right). Write your answer in scientific notation.

Use the commutative property to rearrange the expression.

However 18\times{10^{12}} is not in scientific notation since 18 is not between 1 and 10.

6. Solve \left(9\times{10^7}\right)\div(4\times{10^2}). Write your answer in scientific notation.

Re–write this expression as \cfrac{9\times{10^7}}{4\times{10^2}},

which equals \cfrac{9\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}}{4\times{10}\times{10}}.

Scientific notation FAQs

Yes. For example, Richard, David and James did a survey, asking people what the best genre of movie was. If 1.3\times{10^4} people said sci-fi and 4\times{10^2} more people said historical fiction, how many people said historical fiction? This word problem can be solved by adding the numbers written in scientific notation.

This and other terms, such as ‘standard index form’ or ‘standard form’ (in the UK), all have the same meaning as scientific notation.

These are the digits used to express a number to the desired form of accuracy.

It is a form of number similar to scientific notation, but each number is written so that the exponents of 10 are always multiples of 3.

Each power of 10 has a prefix in the metric system, since the system is based on powers of 10.

The distance a number is from 0.

The next lessons are

Quadratic graphs

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Scientific Notation

Table of Contents

Last modified on August 3rd, 2023

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Scientific notation is a special way of representing numbers which are too large or small in a unique way that makes it easier to remember and compare them. They are expressed in the form (a × 10n). Here ‘a’ is the coefficient, and ‘n’ is the power or exponent of the base 10.

The diagram below shows the standard form of writing numbers in scientific notation:

Thus, scientific notation is a floating-point system where numbers are expressed as products consisting of numbers between 1 and 10 multiplied with appropriate power of 10. It helps to represent big and small numbers in a much easier way.

The speed of light(c) measured in a vacuum is approximately 300,000,000 meters per second, which is written as 3 × 10 8 m/s in scientific notation. Again the mass of the sun is written as 1.988 × 10 30 kg. All these values, if written in scientific form, will reduce a lot of space and decrease the chances of errors.

Scientific Notation Rules

We must follow the five rules when writing numbers in scientific notation:

The base should always be 10
The exponent (n) must be a non-zero integer, positive or negative
The absolute value of the coefficient (a) is greater than or equal to 1, but it should be less than 10 (1 ≤ a < 10)
The coefficient (a) can be positive or negative numbers, including whole numbers and decimal numbers
The mantissa contains the remaining significant digits of the number

How to Do Scientific Notation with Examples

As we know, in scientific notation, there are two parts:

Part 1: Consisting of just the digits with the decimal point placed after the first digit
Part 2: This part follows the first part by × 10 to a power that puts the decimal point where it should be

While writing numbers in scientific notation, we need to figure out how many places we should move the decimal point. The exponent of 10 determines the number of places the decimal point gets shifted to represent the number in long form.

There are two possibilities:

Case 1: With Positive Exponent

When the non-zero digit is followed by a decimal point

For example, if we want to represent 4237.8 in scientific notation, it will be:

The first part will be 4.2378 (only the digit and the decimal point placed after the first digit)
The second part following the first part will be × 10 3 (multiplied by 10 having a power of 3)

Case 2: With Negative Exponent

When the decimal point comes first, and the non-zero digit comes next

For example, if we want to represent 0.000082 in scientific notation, it will be:

The first part will be 8.2 (only the coefficient in decimal form and the decimal point placed after the first digit)
The second part following the first part will be × 10 -5 (multiplied by 10 having a power of -5)

Here is a table showing some more examples of numbers written in scientific notation:

Let us solve some more word problems involving writing numbers in scientific notation.

Write the number 0.0065 in scientific notation.

0.0065 is written in scientific notation as: 6.5 × 10 -3

Convert 4.5 in scientific notation.

4.5 is written in scientific notation as: 4.5 × 10 0

Write 53010000 in scientific notation.

53010000 is written in scientific notation as: 5.301 ×10 7

Light travels with a speed of 1.86 x 10 5 miles/second. It takes sunlight 4.8 x 10 3 seconds to reach Saturn. Find the approximate distance between Sun and Saturn. Express your answer in scientific notation.

As we know, Distance (d) = Speed (s) × Time (t), here s = 1.86 x 10 5 miles/second, t = 4.8 x 10 3 seconds = 8.928 x 10 8 miles

Other Ways of Writing in Scientific Notation

We sometimes use the ^ symbol instead of power while writing numbers in scientific notation. In such cases, the above number 4237.8, written in scientific notation as 4.2378 × 10 3 , can also be written as 4.2378 × 10^3 Similarly, calculators use the notation 4.2378E; here, E signifies 10 × 10 × 10

Converting Scientific Notation to Standard Form
Multiplying Numbers in Scientific Notation
Dividing Numbers in Scientific Notation
Adding and Subtracting Numbers in Scientific Notation

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1.2: Exponents and Scientific Notation

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Learning Objectives

Various rules of Exponents

Scientific Notation

Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is $2,048$ pixels by $1,536$ pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to $48$ bits per frame, and can shoot the equivalent of $24$ frames per second. The maximum possible number of bits of information used to film a one-hour ($3,600$-second) digital film is then an extremely large number.

Using a calculator, we enter $2,048×1$, $536×48×24×3$, $600$ and press ENTER. The calculator displays $1.304596316E13$. What does this mean? The “$E13$” portion of the result represents the exponent $13$ of ten, so there are a maximum of approximately $1.3\times10^{13}$ bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

Using the Product Rule of Exponents

Consider the product $x^3\times x^4$. Both terms have the same base, $x$, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

\[ \begin{align*} x^3 \times x^4 &= \overbrace{x \times x \times x}^{\text{3 factors}} \times \overbrace{ x \times x \times x\times x}^{\text{4 factors}} \\[4pt] &= \overbrace{x\times x\times x\times x\times x\times x\times x}^{\text{7 factors}} \\[4pt] &=x^7 \end{align*}\]

The result is that $x^3\times x^4=x^{3+4}=x^7$.

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

\[a^m\times a^n=a^{m+n}\]

Now consider an example with real numbers.

$2^3\times2^4=2^{3+4}=2^7$

We can always check that this is true by simplifying each exponential expression. We find that $2^3$ is $8$, $2^4$ is $16$, and $2^7$ is $128$. The product $8\times16$ equals $128$, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

THE PRODUCT RULE OF EXPONENTS

For any real number a and natural numbers $m$ and $n$, the product rule of exponents states that

\[a^m\times a^n=a^{m+n} \label{prod}\]

Example $\PageIndex{1}$: Using the Product Rule

Write each of the following products with a single base. Do not simplify further.

$t^5\times t^3$
$(−3)^5\times(−3)$
$x^2\times x^5\times x^3$

Use the product rule (Equation \ref{prod}) to simplify each expression.

$t^5\times t^3=t^{5+3}=t^8$
$(−3)^5\times(−3)=(−3)^5\times(−3)^1=(−3)^{5+1}=(−3)^6$

At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.

\[x^2\times x^5\times x^3=(x^2\times x^5) \times x^3=(x^{2+5})\times x^3=x^7\times x^3=x^{7+3}=x^{10} \nonumber\]

Notice we get the same result by adding the three exponents in one step.

\[x^2\times x^5\times x^3=x^{2+5+3}=x^{10} \nonumber\]

Exercise $\PageIndex{1}$

$k^6\times k^9$
$\left(\dfrac{2}{y}\right)^4\times\left(\dfrac{2}{y}\right)$
$t^3\times t^6\times t^5$

$\left(\dfrac{2}{y}\right)^5$

Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as $\dfrac{y^m}{y^n}$, where $m>n$. Consider the example $\dfrac{y^9}{y^5}$. Perform the division by canceling common factors.

\[\begin{align*} \dfrac{y^9}{y^5} &= \dfrac{y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y}{y\cdot y\cdot y\cdot y\cdot y}\\ &= \dfrac{y\cdot y\cdot y\cdot y}{1}\\ &= y^4 \end{align*}\]

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.

\[\dfrac{a^m}{a^n}=a^{m−n}\]

In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

$\dfrac{y^9}{y^5}=y^{9−5}=y^4$

For the time being, we must be aware of the condition $m>n$. Otherwise, the difference $m-n$ could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

THE QUOTIENT RULE OF EXPONENTS

For any real number $a$ and natural numbers $m$ and $n$, such that $m>n$, the quotient rule of exponents states that

\[\dfrac{a^m}{a^n}=a^{m−n} \label{quot}\]

Example $\PageIndex{2}$: Using the Quotient Rule

$\dfrac{(−2)^{14}}{(−2)^{9}}$
$\dfrac{t^{23}}{t^{15}}$
$\dfrac{(z\sqrt{2})^5}{z\sqrt{2}}$

Use the quotient rule (Equation \ref{quot}) to simplify each expression.

$\dfrac{(−2)^{14}}{(−2)^{9}}=(−2)^{14−9}=(−2)^5$
$\dfrac{t^{23}}{t^{15}}$=t^{23−15}=t^8\)
$\dfrac{(z\sqrt{2})^5}{z\sqrt{2}}=(z\sqrt{2})^{5−1}=(z\sqrt{2})^4$

Exercise $\PageIndex{2}$

$\dfrac{s^{75}}{s^{68}}$
$\dfrac{(−3)^6}{−3}$
$\dfrac{(ef^2)^5}{(ef^2)^3}$

$(−3)^5$

$(ef^2)^2$

Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression $(x^2)^3$. The expression inside the parentheses is multiplied twice because it has an exponent of $2$. Then the result is multiplied three times because the entire expression has an exponent of $3$.

\[\begin{align*} (x^2)^3 &= (x^2)\times(x^2)\times(x^2)\\ &= x\times x\times x\times x\times x\times x\\ &= x^6 \end{align*}\]

The exponent of the answer is the product of the exponents: $(x^2)^3=x^{2⋅3}=x^6$. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.

\[(a^m)^n=a^{m⋅n}\]

Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

THE POWER RULE OF EXPONENTS

For any real number a and positive integers m and n, the power rule of exponents states that

\[(a^m)^n=a^{m⋅n} \label{power}\]

Example $\PageIndex{3}$: Using the Power Rule

$(x^2)^7$
$((2t)^5)^3$
$((−3)^5)^{11}$

Use the power rule (Equation \ref{power}) to simplify each expression.

$(x^2)^7=x^{2⋅7}=x^{14}$
$((2t)^5)^3=(2t)^{5⋅3}=(2t)^{15}$
$((−3)^5)^{11}=(−3)^{5⋅11}=(−3)^{55}$

Exercise $\PageIndex{3}$

$((3y)^8)^3$
$(t^5)^7$
$((−g)^4)^4$

$(3y)^{24}$

$(−g)^{16}$

Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that $m>n$ so that the difference $m−n$ would never be zero or negative. What would happen if $m=n$ ? In this case, we would use the zero exponent rule of exponents to simplify the expression to $1$. To see how this is done, let us begin with an example.

\[\dfrac{t^8}{t^8}=1 \nonumber\]

If we were to simplify the original expression using the quotient rule, we would have

\[\dfrac{t^8}{t^8}=t^{8−8}=t^0 \nonumber\]

If we equate the two answers, the result is $t^0=1$. This is true for any nonzero real number, or any variable representing a real number.

\[a^0=1 \nonumber\]

The sole exception is the expression $0^0$. This appears later in more advanced courses, but for now, we will consider the value to be undefined.

THE ZERO EXPONENT RULE OF EXPONENTS

For any nonzero real number a, the zero exponent rule of exponents states that

Example $\PageIndex{4}$: Using the Zero Exponent Rule

Simplify each expression using the zero exponent rule of exponents.

$\dfrac{c^3}{c^3}$
$\dfrac{-3x^5}{x^5}$
$\dfrac{(j^2k)^4}{(j^2k)\times(j^2k)^3}$
$\dfrac{5(rs^2)^2}{(rs^2)^2}$

Use the zero exponent and other rules to simplify each expression.

a. \[\begin{align*} \dfrac{c^3}{c^3} &= c^{3-3}\\ &= c^0\\ &= 1 \end{align*}\]

b. \[\begin{align*} \dfrac{-3x^5}{x^5} &= -3\times\dfrac{x^5}{x^5}\\ &= -3\times x^{5-5}\\ &= -3\times x^0\\ &= -3\times 1\\ &= -3 \end{align*}\]

c. \[\begin{align*} \dfrac{(j^2k)^4}{(j^2k)\times(j^2k)^3} &= \dfrac{(j^2k)^4}{(j^2k)^{1+3}} && \text{ Use the product rule in the denominator}\\ &= \dfrac{(j^2k)^4}{(j^2k)^4} && \text{ Simplify}\\ &= (j^2k)^{4-4} && \text{ Use the quotient rule}\\ &= (j^2k)^0 && \text{ Simplify}\\ &= 1 \end{align*}\]

d. \[\begin{align*} \dfrac{5(rs^2)^2}{(rs^2)^2} &= 5(rs^2)^{2-2} && \text{ Use the quotient rule}\\ &= 5(rs^2)^0 && \text{ Simplify}\\ &= 5\times1 && \text{ Use the zero exponent rule}\\ &= 5 && \text{ Simplify} \end{align*}\]

Exercise $\PageIndex{4}$

$\dfrac{t^7}{t^7}$
$\dfrac{(de^2)^{11}}{2(de^2)^{11}}$
$\dfrac{w^4\times w^2}{w^6}$
$\dfrac{t^3\times t^4}{t^2\times t^5}$

$\dfrac{1}{2}$

Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that $m>n$ in the quotient rule even further. For example, can we simplify $\dfrac{t^3}{t^5}$? When $m<n$ —that is, where the difference $m−n$ is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example, $\dfrac{t^3}{t^5}$.

\[\begin{align*} \dfrac{t^3}{t^5} &= \dfrac{t\times t\times t}{t\times t\times t\times t\times t} \\ &= \dfrac{1}{t\times t}\\ &= \dfrac{1}{h^2} \end{align*}\]

\[\begin{align*} \dfrac{t^3}{t^5} &= h^{3-5} \\ &= h^{-2} \end{align*}\]

Putting the answers together, we have $h^{−2}=\dfrac{1}{h^2}$. This is true for any nonzero real number, or any variable representing a nonzero real number.

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

We have shown that the exponential expression an is defined when $n$ is a natural number, $0$, or the negative of a natural number. That means that an is defined for any integer $n$. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer $n$.

THE NEGATIVE RULE OF EXPONENTS

For any nonzero real number a and natural number n, the negative rule of exponents states that

\[a^{−n}=\dfrac{1}{a^n}\]

Example $\PageIndex{5}$: Using the Negative Exponent Rule

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

$\dfrac{\theta^3}{\theta^{10}}$
$\dfrac{z^2\times z}{z^4}$
$\dfrac{(-5t^3)^4}{(-5t^3)^8}$
$\dfrac{\theta^3}{\theta^{10}}=\theta^{3-10}=\theta^{-7}=\dfrac{1}{\theta^7}$
$\dfrac{z^2\times z}{z^4}=\dfrac{z^{2+1}}{z^4}=\dfrac{z^3}{z^4}=z^{3-4}=z^{-1}=\dfrac{1}{z}$
$\dfrac{(-5t^3)^4}{(-5t^3)^8}=(-5t^3)^{4-8}=(-5t^3)^{-4}=\dfrac{1}{(-5t^3)^4}$

Exercise $\PageIndex{5}$

$\dfrac{(-3t)^2}{(-3t)^8}$
$\dfrac{f^{47}}{f^{49}\times f}$
$\dfrac{2k^4}{5k^7}$

$\dfrac{1}{(-3t)^6}$

$\dfrac{1}{f^3}$

$\dfrac{2}{5k^3}$

Example $\PageIndex{6}$: Using the Product and Quotient Rules

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

$b^2\times b^{-8}$
$(-x)^5\times(-x)^{-5}$
$\dfrac{-7z}{(-7z)^5}$
$b^2\times b^{-8}=b^{2-8}=b^{-6}=\dfrac{1}{b^6}$
$(-x)^5\times(-x)^{-5}=(-x)^{5-5}=(-x)^0=1$
$\dfrac{-7z}{(-7z)^5}= \dfrac{(-7z)^1}{(-7z)^5}=(-7z)^{1-5}=(-7z)^{-4}=\dfrac{1}{(-7z)^4}$

Exercise $\PageIndex{6}$

$t^{-11}\times t^6$
$\dfrac{25^{12}}{25^{13}}$

$t^{-5}=\dfrac{1}{t^5}$

$\dfrac{1}{25}$

Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider $(pq)^3$. We begin by using the associative and commutative properties of multiplication to regroup the factors.

\[\begin{align*} (pq)^3 &= (pq)\times(pq)\times(pq)\\ &= p\times q\times p\times q\times p\times q\\ &= p^3\times q^3 \end{align*}\]

In other words, $(pq)^3=p^3\times q^3$.

THE POWER OF A PRODUCT RULE OF EXPONENTS

For any real numbers a and b and any integer n, the power of a product rule of exponents states that

\[(ab)^n=a^nb^n\]

Example $\PageIndex{7}$: Using the Power of a Product Rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

$(ab^2)^3$
$(2t)^{15}$
$(-2w^3)^3$
$\dfrac{1}{(-7z)^4}$
$(e^{-2}f^2)^7$

Use the product and quotient rules and the new definitions to simplify each expression.

a. $(ab^2)^3=(a)^3\times(b^2)^3=a^{1\times3}\times b^{2\times3}=a^3b^6$

b. $(2t)^{15}=(2)^{15}\times(t)^{15}=2^{15}t^{15}=32,768t^{15}$

c. $(−2w^3)^3=(−2)^3\times(w^3)^3=−8\times w^{3\times3}=−8w^9$

d. $\dfrac{1}{(-7z)^4}=\dfrac{1}{(-7)^4\times(z)^4}=\dfrac{1}{2401z^4}$

e. $(e^{-2}f^2)^7=(e^{−2})^7\times(f^2)^7=e^{−2\times7}\times f^{2\times7}=e^{−14}f^{14}=\dfrac{f^{14}}{e^{14}}$

Exercise $\PageIndex{7}$

$(g^2h^3)^5$
$(-3y^5)^3$
$\dfrac{1}{(a^6b^7)^3}$
$(r^3s^{-2})^4$

$g^{10}h^{15}$

$-27y^{15}$

$\dfrac{1}{a^{18}b^{21}}$

$\dfrac{r^{12}}{s^8}$

Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

\[(e^{−2}f^2)^7=\dfrac{f^{14}}{e^{14}}\]

Let’s rewrite the original problem differently and look at the result.

\[\begin{align*} (e^{-2}f^2)^7 &= \left(\dfrac{f^2}{e^2}\right)^7\\ &= \dfrac{f^{14}}{e^{14}} \end{align*}\]

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

\[\begin{align*} (e^{-2}f^2)^7 &= \left(\dfrac{f^2}{e^2}\right)^7\\ &= \dfrac{(f^2)^7}{(e^2)^7}\\ &= \dfrac{f^{2\times7}}{e^{2\times7}}\\ &= \dfrac{f^{14}}{e^{14}} \end{align*}\]

THE POWER OF A QUOTIENT RULE OF EXPONENTS

For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that

\[\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\]

Example $\PageIndex{8}$: Using the Power of a Quotient Rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

$\left(\dfrac{4}{z^{11}}\right)^3$
$\left(\dfrac{p}{q^3}\right)^6$
$\left(\dfrac{-1}{t^2}\right)^{27}$
$(j^3k^{-2})^4$
$(m^{-2}n^{-2})^3$

a. $\left(\dfrac{4}{z^{11}}\right)^3=\dfrac{(4)^3}{(z^{11})^3}=\dfrac{64}{z^{11\times3}}=\dfrac{64}{z^{33}}$

b. $\left(\dfrac{p}{q^3}\right)^6=\dfrac{(p)^6}{(q^3)^6}=\dfrac{p^{1\times6}}{q^{3\times6}}=\dfrac{p^6}{q^{18}}$

c. $\left(\dfrac{-1}{t^2}\right)^{27}=\dfrac{(-1)^{27}}{(t^2)^{27}}=\dfrac{-1}{t^{2\times27}}=\dfrac{-1}{t^{54}}=-\dfrac{1}{t^{54}}$

d. $(j^3k^{-2})^4=\left(\dfrac{j^3}{k^2}\right)^4=\dfrac{(j^3)^4}{(k^2)^4}=\dfrac{j^{3\times4}}{k^{2\times4}}=\dfrac{j^{12}}{k^8}$

e. $(m^{-2}n^{-2})^3=\left(\dfrac{1}{m^2n^2}\right)^3=\dfrac{(1)^3}{(m^2n^2)^3}=\dfrac{1}{(m^2)^3(n^2)^3}=\dfrac{1}{m^{2\times3}n^{2\times3}}=\dfrac{1}{m^6n^6}$

Exercise $\PageIndex{8}$

$\left(\dfrac{b^5}{c}\right)^3$
$\left(\dfrac{5}{u^8}\right)^4$
$\left(\dfrac{-1}{w^3}\right)^{35}$
$(p^{-4}q^3)^8$
$(c^{-5}d^{-3})^4$

$\dfrac{b^{15}}{c^3}$

$\dfrac{625}{u^{32}}$

$\dfrac{-1}{w^{105}}$

$\dfrac{q^{24}}{p^{32}}$

$\dfrac{1}{c^{20}d^{12}}$

Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Example $\PageIndex{9}$: Simplifying Exponential Expressions

Simplify each expression and write the answer with positive exponents only.

$(6m^2n^{-1})^3$
$17^5\times17^{-4}\times17^{-3}$
$\left(\dfrac{u^{-1}v}{v^{-1}}\right)^2$
$(-2a^3b^{-1})(5a^{-2}b^2)$
$(x^2\sqrt{2})^4(x^2\sqrt{2})^{-4}$
$\dfrac{(3w^2)^5}{(6w^{-2})^2}$

a. \[\begin{align*} (6m^2n^{-1})^3 &= (6)^3(m^2)^3(n^{-1})^3 && \text{ The power of a product rule}\\ &= 6^3m^{2\times3}n^{-1\times3} && \text{ The power rule}\\ &= 216m^6n^{-3} && \text{ The power rule}\\ &= \dfrac{216m^6}{n^3} && \text{ The negative exponent rule} \end{align*}\]

b. \[\begin{align*} 17^5\times17^{-4}\times17^{-3} &= 17^{5-4-3} && \text{ The product rule}\\ &= 17^{-2} && \text{ Simplify}\\ &= \dfrac{1}{17^2} \text{ or } \dfrac{1}{289} && \text{ The negative exponent rule} \end{align*}\]

c. \[\begin{align*} \left ( \dfrac{u^{-1}v}{v^{-1}} \right )^2 &= \dfrac{(u^{-1}v)^2}{(v^{-1})^2} && \text{ The power of a quotient rule}\\ &= \dfrac{u^{-2}v^2}{v^{-2}} && \text{ The power of a product rule}\\ &= u^{-2}v^{2-(-2)} && \text{ The quotient rule}\\ &= u^{-2}v^4 && \text{ Simplify}\\ &= \dfrac{v^4}{u^2} && \text{ The negative exponent rule} \end{align*}\]

d. \[\begin{align*} \left (-2a^3b^{-1} \right ) \left(5a^{-2}b^2 \right ) &= \left (x^2\sqrt{2} \right )^{4-4} && \text{ Commutative and associative laws of multiplication}\\ &= -10\times a^{3-2}\times b^{-1+2} && \text{ The product rule}\\ &= -10ab && \text{ Simplify} \end{align*}\]

e. \[\begin{align*} \left (x^2\sqrt{2})^4(x^2\sqrt{2} \right )^{-4} &= \left (x^2\sqrt{2} \right )^{4-4} && \text{ The product rule}\\ &= \left (x^2\sqrt{2} \right )^0 && \text{ Simplify}\\ &= 1 && \text{ The zero exponent rule} \end{align*}\]

f. \[\begin{align*} \dfrac{(3w^2)^5}{(6w^{-2})^2} &= \dfrac{(3)^5\times(w^2)^5}{(6)^2\times(w^{-2})^2} && \text{ The power of a product rule}\\ &= \dfrac{3^5w^{2\times5}}{6^2w^{-2\times2}} && \text{ The power rule}\\ &= \dfrac{243w^{10}}{36w^{-4}} && \text{ Simplify}\\ &= \dfrac{27w^{10-(-4)}}{4} && \text{ The quotient rule and reduce fraction}\\ &= \dfrac{27w^{14}}{4} && \text{ Simplify} \end{align*}\]

Using Scientific Notation

Recall at the beginning of the section that we found the number $1.3\times10^{13}$ when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about $0.00005\; m$, and the radius of an electron, which is about $0.00000000000047\; m$. How can we effectively work read, compare, and calculate with numbers such as these?

A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of $10$. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $1$ and $10$. Count the number of places $n$ that you moved the decimal point. Multiply the decimal number by $10$ raised to a power of $n$. If you moved the decimal left as in a very large number, $n$ is positive. If you moved the decimal right as in a small large number, $n$ is negative.

For example, consider the number $2,780,418$. Move the decimal left until it is to the right of the first nonzero digit, which is $2$.

$Arrows representing moving the decimal point.$

We obtain $2.780418$ by moving the decimal point $6$ places to the left. Therefore, the exponent of $10$ is $6$, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

Working with small numbers is similar. Take, for example, the radius of an electron, $0.00000000000047\; m$. Perform the same series of steps as above, except move the decimal point to the right.

$Arrows representing moving the decimal point.$

Be careful not to include the leading $0$ in your count. We move the decimal point $13$ places to the right, so the exponent of $10$ is $13$. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

SCIENTIFIC NOTATION

A number is written in scientific notation if it is written in the form $a\times{10}^n$, where $1≤|a|<10$ and $n$ is an integer.

Example $\PageIndex{10}$: Converting Standard Notation to Scientific Notation

Write each number in scientific notation.

Distance to Andromeda Galaxy from Earth: $24,000,000,000,000,000,000,000\; m$
Diameter of Andromeda Galaxy: $1,300,000,000,000,000,000,000\; m$
Number of stars in Andromeda Galaxy: $1,000,000,000,000$
Diameter of electron: $0.00000000000094\; m$
Probability of being struck by lightning in any single year: $0.00000143$

a. $24,000,000,000,000,000,000,000\; m$ $22$ places

$2.4\times{10}^{22}\; m$

b. $1,300,000,000,000,000,000,000\; m$ $21$ places

$1.3\times{10}^{21}\; m$

c. $1,000,000,000,000$ $12$ places

$1\times{10}^{12}$

d. $0.00000000000094\; m$ $13$ places

$9.4\times{10}^{-13}\; m$

e. $0.00000143$ $6$ places

$1.43\times{10}^6$

Observe that, if the given number is greater than $1$, as in examples a–c, the exponent of $10$ is positive; and if the number is less than $1$, as in examples d–e, the exponent is negative.

Exercise $\PageIndex{10}$

U.S. national debt per taxpayer (April 2014): $\$152,000$
World population (April 2014): $7,158,000,000$
World gross national income (April 2014): $\$85,500,000,000,000$
Time for light to travel $1\; m: 0.00000000334\; s$
Probability of winning lottery (match $6$ of $49$ possible numbers): $0.0000000715$

$\$1.52\times{10}^5$

$7.158\times{10}^9$

$\$8.55\times{10}^{13}$

$3.34\times{10}^{-9}$

$7.15\times{10}^{-8}$

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal n places to the right if $n$ is positive or $n$ places to the left if $n$ is negative and add zeros as needed. Remember, if $n$ is positive, the value of the number is greater than $1$, and if $n$ is negative, the value of the number is less than one.

Example $\PageIndex{11}$: Converting Scientific Notation to Standard Notation

Convert each number in scientific notation to standard notation.

$3.547\times{10}^{14}$
$−2\times{10}^6$
$7.91\times{10}^{−7}$
$−8.05\times{10}^{−12}$

a. $3.547\times{10}^{14}$

$3.54700000000000$

$\rightarrow14$ places

$354,700,000,000,000$

b. $−2\times{10}^6$

$−2.000000$

$\rightarrow6$ places

$−2,000,000$

c. $7.91\times{10}^{−7}$

$0000007.91$

$\rightarrow7$ places

$0.000000791$

d. $−8.05\times{10}^{−12}$

$−000000000008.05$

$\rightarrow12$ places

$−0.00000000000805$

Exercise $\PageIndex{11}$

$7.03\times{10}^5$
$−8.16\times{10}^{11}$
$−3.9\times{10}^{−13}$
$8\times{10}^{−6}$

$703,000$

$−816,000,000,000$

$−0.00000000000039$

$0.000008$

Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in $1\; L$ of water. Each water molecule contains $3$ atoms ($2$ hydrogen and $1$ oxygen). The average drop of water contains around $1.32\times{10}{21}$ molecules of water and $1\; L$ of water holds about $1.22\times{10}^{4}$ average drops. Therefore, there are approximately $3⋅(1.32\times{10}^{21})⋅(1.22\times{10}^4)≈4.83\times{10}^{25}$ atoms in $1\; L$ of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product $(7\times{10}^4)⋅(5\times{10}^6)=35\times{10}^{10}$. The answer is not in proper scientific notation because $35$ is greater than $10$. Consider $35$ as $3.5\times10$. That adds a ten to the exponent of the answer.

$(35)\times{10}^{10}=(3.5\times10)\times{10}^{10}=3.5\times(10\times{10}^{10})=3.5\times{10}^{11}$

Example $\PageIndex{12}$: Using Scientific Notation

Perform the operations and write the answer in scientific notation.

$(8.14\times{10}^{−7})(6.5\times{10}^{10})$
$(4\times{10}^5)÷(−1.52\times{10}^{9})$
$(2.7\times{10}^5)(6.04\times{10}^{13})$
$(1.2\times{10}^8)÷(9.6\times{10}^5)$
$(3.33\times{10}^4)(−1.05\times{10}^7)(5.62\times{10}^5)$

a. \[\begin{align*} (8.14\times{10}^{-7})(6.5\times{10}^{10}) &= (8.14\times6.5)({10}^{-7}\times{10}^{10}) \text{ Commutative and associative properties of multiplication}\\ &= (52.91)({10}^3) \text{ Product rule of exponents}\\ &= 5.291\times{10}^4 \text{ Scientific notation} \end{align*}\]

b. \[\begin{align*} (4\times{10}^5)\div (-1.52\times{10}^{9}) &= \left(\dfrac{4}{-1.52}\right)\left(\dfrac{{10}^5}{{10}^9}\right) \text{ Commutative and associative properties of multiplication}\\ &\approx (-2.63)({10}^{-4}) \text{ Quotient rule of exponents}\\ &= -2.63\times{10}^{-4} \text{ Scientific notation} \end{align*}\]

c. \[\begin{align*} (2.7\times{10}^5)(6.04\times{10}^{13}) &= (2.7\times6.04)({10}^5\times{10}^{13}) \text{ Commutative and associative properties of multiplication}\\ &= (16.308)({10}^{18}) \text{ Product rule of exponents}\\ &= 1.6308\times{10}^{19} \text{ Scientific notation} \end{align*}\]

d. \[\begin{align*} (1.2\times{10}^8)÷(9.6\times{10}^5) &= \left(\dfrac{1.2}{9.6}\right)\left(\dfrac{{10}^8}{{10}^5}\right) \text{ Commutative and associative properties of multiplication}\\ &= (0.125)({10}^3) \text{ Quotient rule of exponents}\\ &= 1.25\times{10}^2 \text{ Scientific notation} \end{align*}\]

e. \[\begin{align*} (3.33\times{10}^4)(-1.05\times{10}^7)(5.62\times{10}^5) &= [3.33\times(-1.05)\times5.62]({10}^4\times{10}^7\times{10}^5)\\ &\approx (-19.65)({10}^{16})\\ &= -1.965\times{10}^{17} \end{align*}\]

Exercise $\PageIndex{12}$

$(−7.5\times{10}^8)(1.13\times{10}^{−2})$
$(1.24\times{10}^{11})÷(1.55\times{10}^{18})$
$(3.72\times{10}^9)(8\times{10}^3)$
$(9.933\times{10}^{23})÷(−2.31\times{10}^{17})$
$(−6.04\times{10}^9)(7.3\times{10}^2)(−2.81\times{10}^2)$

$−8.475\times{10}^6$

$8\times{10}^{−8}$

$2.976\times{10}^{13}$

$−4.3\times{10}^6$

$≈1.24\times{10}^{15}$

Example $\PageIndex{13}$: Applying Scientific Notation to Solve Problems

In April 2014, the population of the United States was about $308,000,000$ people. The national debt was about $\$17,547,000,000,000$. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.

The population was $308,000,000=3.08\times{10}^8$.

The national debt was $$17,547,000,000,000≈$1.75\times{10}^{13}$.

To find the amount of debt per citizen, divide the national debt by the number of citizens.

\[\begin{align*} (1.75\times{10}^{13})\div (3.08\times{10}^8)&=\left(\dfrac{1.75}{3.08}\right)({10}^5)\\ &\approx 0.57\times{10}^5\\ &=5.7\times{10}^4 \end{align*}\]

The debt per citizen at the time was about $$5.7\times{10}^4$, or $$57,000$.

Exercise $\PageIndex{13}$

An average human body contains around $30,000,000,000,000$ red blood cells. Each cell measures approximately $0.000008\; m$ long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

Number of cells: $3\times{10}^{13}$; length of a cell: $8\times{10}^{−6}\; m$; total length: $2.4\times{10}^8\; m$ or $240,000,000\; m$.

Access these online resources for additional instruction and practice with exponents and scientific notation.

Exponential Notation

Properties of Exponents

Zero Exponent

Simplify Exponent Expressions

Quotient Rule for Exponents

Converting to Decimal Notation

Key Equations

Key concepts.

Products of exponential expressions with the same base can be simplified by adding exponents. See Example .
Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example .
Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example .
An expression with exponent zero is defined as 1. See Example .
An expression with a negative exponent is defined as a reciprocal. See Example and Example .
The power of a product of factors is the same as the product of the powers of the same factors. See Example .
The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example .
The rules for exponential expressions can be combined to simplify more complicated expressions. See Example .
Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example and Example .
Scientific notation may be used to simplify calculations with very large or very small numbers. See Example and Example .

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HiSET: Math : Solve problems using scientific notation

Study concepts, example questions & explanations for hiset: math, all hiset: math resources, example questions, example question #1 : solve problems using scientific notation.

Simplify the following expression using scientific notation.