Have you ever tried to write a number in scientific notation? Evan has a dilemma. Take a look.

Evan is trying to work on his math homework. He is faced with a dilemma where he is being asked to write a number in scientific notation. Here is the number.

\begin{align*}9.87e-7\end{align*}

Evan isn't sure how to do this. Do you know?

**This Concept is all about scientific notation. By the end of it, you will know how to help Evan with this dilemma.**

### Guidance

**What is scientific notation?**

** Scientific Notation** is a shortcut for writing very small and very large numbers.

When you write in scientific notation, you write a number between 1 and 10 multiplied by a power of ten. Here is an example of a number and the same number written in scientific notation:

\begin{align*} 450,000 = 4.5\times 10\times 10\times 10\times 10\times 10= 4.5 \times 10^5\end{align*}

\begin{align*}4.5\times 10^5\end{align*} is scientific notation. Large numbers written using scientific notation will use positive exponents. Note that to change 450,000 into 4.5, you must move the decimal point five spaces to the left. This is why when the number is written in scientific notation the exponent is 5.

**What about very small numbers written using scientific notation?**

\begin{align*}.0023 = 2.3 \div 10\div 10 \div 10 = 2.3 \times 10^{-3}\end{align*}

\begin{align*}2.3\times 10^{-3}\end{align*} is scientific notation. Multiplying by \begin{align*}10^{-3}\end{align*} is like dividing by 10 three times. When writing small numbers between 0 and 1 using scientific notation, we will use negative exponents. Note that to change .0023 into 2.3, you must move the decimal point three spaces to the right. This is why when the number is written in scientific notation the exponent is -3.

.00056

**If we want to write this in scientific notation, we first start with the number between 1 and 10. This number is 5.6.**

**5.6 \begin{align*}\times\end{align*} _____**

**We want to multiply 5.6 by a power of ten. Since .00056 is a number less than 1, we know that it will be a negative power of ten. Notice that to go from .00056 to 5.6, you must move the decimal point four places to the right. This means the exponent will be \begin{align*}-4\end{align*}.**

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

**We can work the other way around too. If we have the scientific notation, we can write the original number by moving the decimal point. If the exponent is negative, work backwards and move the decimal point to the left. If the exponent is positive, work backwards and move the decimal point to the right. Move the decimal point the number of times indicated by the exponent.**

\begin{align*}3.2 \times 10^{-5} = .000032\end{align*}

Notice that to determine the original number, we moved the decimal point five times to the left.

**Scientific notation is very useful for scientists, mathematicians and engineers. It is useful in careers where people work with very large or very small numbers.**

Practice writing these numbers in scientific notation.

#### Example A

.0012 = _____

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

#### Example B

**78,000,000 = _____**

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

#### Example C

**345,102,000,000 = _____**

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

Now back to Evan. Here is the original problem once again.

Evan is trying to work on his math homework. He is faced with a dilemma where he is being asked to write a number in scientific notation. Here is the number.

\begin{align*}9.87e-7\end{align*}

To write this in scientific notation, we first need to look at which way we are going to move the decimal point. Because this is a very tiny decimal, we are going to move the decimal point to the right. We are going to move it 7 places.

\begin{align*}9.87\end{align*}

But wait a minute! We aren't done yet. We have to add in the power to show how many places we moved the decimal point.

\begin{align*}9.87 \times 10^{-7}\end{align*}

**This is our answer.**

### Vocabulary

- Power of 10
- \begin{align*}10^{1}, 10^{2}, 10^{3}, \cdots \end{align*} and \begin{align*}10^{-1}, 10^{-2}, 10^{-3},\cdots \end{align*}.

- Scientific notation
- A means of representing a number as a product of a number between 1 and 10 and a power of 10.

### Guided Practice

Here is one for you to try on your own.

Write the following number in scientific notation.

\begin{align*}3.4e-9\end{align*}

**Answer**

First, we are going to move the decimal point 9 places to the right.

\begin{align*}3.4\end{align*}

Next, we add in the power. Notice that the exponent is negative because we moved the decimal to the right.

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

**This is our answer.**

### Video Review

James Sousa Dividing by Powers of Ten

Khan Academy: Scientific Notation Examples

### Practice

Directions: Write each decimal in scientific notation.

1. .00045

2. .098

3. 30,000,000

4. .000987

5. 3,400,000

6. .0000021

7. 1,230,000,000,000

8. .00000000345

9. .00056

10. .0098

11. .024

12. .000023

13. 4,300

14. .0000000000128

15. 980

16. .00000045