Introduction
Have you ever been curious about the size of different planets in space? At the science museum, Hailey and Aaron are very interested in learning more about just these types of things.
Hailey and Aaron are very interested in Astronomy, so they were very excited when their group reached the Astronomy exhibit. Aaron 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. Hailey 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 Hailey 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 Hailey 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.
Guided Practice
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: _____
Example B
34.567 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} _____
Solution: _____
Example C
127.3 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} _____
Solution: _____
Now that you understand multiplying by powers of ten, you can start by helping Hailey answer her questions. To figure out the diameter or the distance across the earth, Hailey 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!
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 _____.
Video Review
Here is a video for review.
James Sousa: Multiplying by Powers of Ten
Practice Set
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. Complete the table.
89.761 | x 10 = | |
89.761 | x 100 = | |
89.761 | x 1000 = |
17. Complete the table.
3.456 | x 10 = | |
3.456 | x 100 = | |
3.456 | x 1000 = |
Review
- We move the decimal point to the right when we multiply by multiples of ten.
- We move the decimal point the same number of places as there are zeros.