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# Scientific Notation Values

## Decimals written as a power of ten

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Scientific Notation Values

David is working on his science homework. The assignment is to find the mass of each planet in the solar system and write it using scientific notation. While researching the assignment, he finds that the mass of the Earth is 5,973,600,000,000,000,000,000,000 kg. How can David express the mass of the Earth in scientific notation?

In this concept, you will learn how to write numbers using scientific notation.

### Scientific Notation

Scientific notation is a shortcut for writing very small and very large numbers. It is very useful for scientists, mathematicians, and engineers. It is useful in careers where people work with very large or very small numbers. For example, the distance from the Earth to the Sun is 96,960,000 miles. Instead of writing out the number every time, you can use scientific notation. Numbers in scientific notation follow the form

\begin{align*}a \times 10^n,\end{align*}

where \begin{align*}a\end{align*}, a number between 1 and 10, is multiplied by a power of ten.

Remember that a power of ten is written as \begin{align*}10^n\end{align*}, where \begin{align*}n\end{align*} is the exponent that tells you how many times 10 is multiplied by itself.

Here is 96,960,000 written in scientific notation.

\begin{align*}96,960,000=9.696 \times 10^7\end{align*}

To change 96,960,000 into a number between 1 and 10, you must move the decimal point 7 spaces to the left. So to get 96,960,000 from 9.696, you must do the opposite and move the decimal point 7 spaces to the right. Moving the decimal point to the right requires multiplying by 10 a total of 7 times or \begin{align*}10^7\end{align*}. Note that large numbers written in scientific notation will use positive exponents.

Here is a very small number written in scientific notation.

\begin{align*}0.000000023 = 2.3 \times 10^{-8}\end{align*}

To change 0.000000023 into a number between 1 and 10, you must move the decimal point 8 spaces to the right. So to convert 2.3 to 0.000000023, move the decimal 8 spaces to the left. Moving the decimal to the left requires multiplying by a negative power of 10 a total of 8 times or \begin{align*}10^{-8}\end{align*}. Remember that multiplying by a negative power of ten is the same as dividing by a power of ten.

\begin{align*}10^{-8} = \frac{1}{10^8}\end{align*}

Note that decimal numbers less than 1 will use negative powers of ten when written in scientific notation.

Here is a small decimal number. Write the number using scientific notation.

\begin{align*}0.00056\end{align*}

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 5.6.

\begin{align*}5.6 \times 10^{n}\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since 0.00056 is a number less than 1, the power of ten will be negative. Notice that to go from 0.00056 to 5.6, you must move the decimal point four places to the right. This means the exponent will be −4.

\begin{align*}5.6 \times 10^{-4}\end{align*}

To check if this is correct, multiply 5.6 times \begin{align*}10^{-4}\end{align*}. When multiplying by a negative power of ten, move the decimal point to the left 4 times.

\begin{align*}0.\underleftarrow{0005}6\end{align*}

Here are some charts that might help you remember how to convert numbers to scientific notation and scientific notation to numbers.

Converting a Number to Scientific Notation

Large Numbers → Positive Power of Ten

Small Decimal Numbers → Negative Power of Ten

Converting Scientific Notation to a Number

Positive Power of Ten → Move the Decimal to the Right

Negative Power of Ten → Move the Decimal to the Left

### Examples

#### Example 1

Earlier, you were given a problem about David’s science homework.

David needs to express the mass on the Earth, 5,973,600,000,000,000,000,000,000 kg, using scientific notation.

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 5.9736.

\begin{align*}5.9736 \times 10^n\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since the mass of the Earth is a large number, the power of ten will be positive. The decimal moves 24 places to get from 5,973,600,000,000,000,000,000,000 to 5.9736, so \begin{align*}n\end{align*} is 24.

\begin{align*}5.9736 \times 10^{24}\end{align*}

The scientific notation form of 5,973,600,000,000,000,000,000,000 kg is \begin{align*}5.9736 \times 10^{24} \ kg\end{align*}.

#### Example 2

Write the number in scientific notation.

0.0000000034

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 3.4.

\begin{align*}3.4 \times 10^n\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since 0.0000000034 is a small number, the power of ten will be negative. The decimal moves 9 places to get from 0.0000000034 to 3.4, so \begin{align*}n\end{align*} is -9.

\begin{align*}3.4 \times 10^{-9}\end{align*}

The scientific notation form of 0.0000000034 is \begin{align*}3.4 \times 10^{-9}\end{align*}.

Write the numbers in scientific notation.

#### Example 3

\begin{align*}0.0012\end{align*}

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 1.2.

\begin{align*}1.2 \times 10^n\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since 0.0012 is a small number, the power of ten will be negative. The decimal moves 3 places to get from 0.0012 to 1.2, so \begin{align*}n\end{align*} is -3.

\begin{align*}1.2 \times 10^{-3}\end{align*}

The scientific notation form of 0.0012 is \begin{align*}1.2 \times 10^{-3}\end{align*}.

#### Example 4

\begin{align*}78,000,000\end{align*}

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 7.8.

\begin{align*}7.8 \times 10^n\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since 78,000,000 is a large number, the power of ten will be positive. The decimal moves 7 places to get from 78,000,000 to 7.8, so \begin{align*}n\end{align*} is 7.

\begin{align*}7.8 \times 10^7\end{align*}

The scientific notation form of 78,000,000 is \begin{align*}7.8 \times 10^7\end{align*}.

#### Example 5

\begin{align*}345,102,000,000\end{align*}

First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 3.45102.

\begin{align*}3.45102 \times 10^n\end{align*}

Then, identify the power of ten, \begin{align*}n\end{align*}. Since 345,102,000,000 is a large number, the power of ten will be positive. The decimal moves 11 places to get from 345,102,000,000 to 3.45102, so \begin{align*}n\end{align*} is 11.

\begin{align*}3.45102 \times 10^{11}\end{align*}

The scientific notation form of 345,102,000,000 is \begin{align*}3.45102 \times 10^{11}\end{align*}.

### Review

Write each decimal in scientific notation.

1. 0.00045
2. 0.098
3. 30,000,000
4. 0.000987
5. 3,400,000
6. 0.0000021
7. 1,230,000,000,000
8. 0.00000000345
9. 0.00056
10. 0.0098
11. 0.024
12. 0.000023
13. 4,300
14. 0.0000000000128
15. 980
16. 0.00000045

To see the Review answers, open this PDF file and look for section 4.15.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Scientific Notation

Scientific notation is a means of representing a number as a product of a number that is at least 1 but less than 10 and a power of 10.