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 numbers and decimals.

When you write in scientific notation, you write decimals times the power of ten that the decimal was multiplied by. You could think of scientific notation as working backwards from multiplying decimals by powers of ten.

\begin{align*} 4500 = 45 \times 10^2\end{align*}

This problem has a whole number and not a decimal. We start with a number called 4500, this has two decimal places in it. Therefore, we are going to say that if we multiplied 45 by 10 squared, we would have 4500 as our number.

**Whole number scientific notation has positive exponents. What about decimal scientific notation?**

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

**What does this mean?**

It means that to write the decimal, we had to multiply this decimal by a power of ten that is negative because our decimal had to move three places to the right to become a whole number with additional decimal places. When we write a decimal in scientific notation, we use negative exponents. Our number isn’t negative, but the direction that we move the decimal point is represented by negative exponents.

.00056

**If we want to write this in scientific notation, we first start with the decimal. This decimal becomes 5.6.**

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

**We want to multiply 5.6 by a power of ten. Since this is a decimal, we know that it will be a negative power of ten. Since we moved the decimal point four places, it will be a negative four exponent.**

\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 decimal.**

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

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

Practice writing a few of these decimals in scientific notation.

#### Example A

.0012 = _____

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

#### Example B

**.00078 = _____**

**Solution:** \begin{align*}7.8 \times 10^{-4}\end{align*}

#### Example C

**.0000023 = _____**

**Solution:** \begin{align*}2.3 \times 10^{-6}\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 ten
- 10, 100, 1000, 10,000 - you can think of them as multiples of ten.

- Scientific notation
- a way to write decimals and numbers by writing a number sentence that shows a power of ten using an exponent.

**Remember:**

Multiplying by a power of ten with a positive exponent means the decimal point was moved to the right.

Multiplying by a power of ten with a negative exponent means the decimal point was moved to the left.

### Guided Practice

Here is one for you to try on your own.

Write the following decimal 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. .00003

4. .000987

5. .000034

6. .0000021

7. .000000123

8. .00000000345

9. .00056

10. .0098

11. .024

12. .000023

13. .00000043

14. .0000000000128

15. .00000000000098

16. .00000045