Have you ever been curious about the size of different planets in space? At the science museum, Kailey and Aron are very interested in learning more about just these types of things.

Kailey and Aron are very interested in Astronomy, so they were very excited when their group reached the Astronomy exhibit. Aron is particularly interested in how fast you can travel from the earth to the moon and to other planets. He found an interactive activity on figuring this out and was very excited. Kailey gravitated over to an interactive exhibit about the earth. In this exhibit, the students are required to figure out what would happen if the size of the earth were increased or decreased. The diameter of the earth is 12,756.3 km. As Kailey starts to work on the activity, she starts off with the first question.

What would the diameter of the earth be if it were ten times as large?

As Kailey starts to think about this question, she realizes that she is going to need to multiply a very large number by ten. She is sure that she can do it in her head, but isn't sure how.

Do you know?

**In this Concept, you will learn how complete problems just like this one.**

### Guidance

This Concept involves a lot of mental math, so try to work without a piece of paper and a pencil as we go through this. You have already learned how to multiply decimals by whole numbers, however, there is a pattern that you can follow when you multiply decimals by whole number *powers of ten.*

**What is the pattern when I multiply decimals by whole number powers of ten?**

\begin{align*}3.4 \times 10 & = 34\\ 3.45 \times 100 & = 345\\ .367 \times 10 & = 3.67\\ .45 \times 1000 & = 450\end{align*}

**If you look carefully you will see that we move the decimal point to the right when we multiply by multiples of ten.**

**How many places do we move the decimal point?**

**That depends on the base ten number. An easy way to think about it is that you move the decimal point the same number of places as there are zeros.**

If you look at the first example, ten has one zero and the decimal point moved one place to the **right**. In the second example, one hundred has two zeros and the decimal point moved two places to the **right**.

You get the idea.

Now it is your turn to practice. Use mental math to multiply each decimal and multiple of ten.

#### Example A

.23 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} _____

**Solution: 2.3**

#### Example B

34.567 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} _____

**Solution: 3456.7**

#### Example C

127.3 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} _____

**Solution: 1,273**

Now that you understand multiplying by powers of ten, you can start by helping Kailey answer her questions. To figure out the diameter or the distance across the earth, Kailey has to use multiplication and division by powers of ten.

She knows that the diameter of the earth is 12,756.3 km. If it were 10 times as large, she would multiply this number by 10. Remember that when you multiply by a whole number power of ten, you move the decimal point one place to the right.

**12,756.3 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} 127,563 km**

**Wow! That is some difference in size!**

### Vocabulary

- Power of ten
- 10, 100, 1000, 10,000 - you can think of them as multiples of ten.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}4.567 \times 1000\end{align*} = _____

**Answer**

We are going to multiply by 1000, so we move the decimal point three places to the right.

**Our answer is \begin{align*}4,567\end{align*}.**

### Video Review

James Sousa: Multiplying by Powers of Ten

### Practice

Directions: Use mental math to multiply each decimal by a whole number power of ten.

1. 3.4 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

2. 3.45 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

3. .56 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

4. 1.234 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} ______

5. 87.9 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

6. 98.32 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

7. 7.2 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} ______

8. 12.5 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

9. 18.91 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

10. 16.57 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} ______

11. 3.44 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

12. .3467 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

13. 7.89 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

14. .3402 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} ______

15. .0123 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} ______

16. .003456 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} ______

17. .89761 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} ______