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# Mental Math to Multiply by Decimal Powers of Ten

## Find products of decimal powers mentally.

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Mental Math to Multiply by Decimal Powers of Ten

Have you ever thought about asteroids?

Aron is learning all about them at the science museum. In this activity, Aron is asked to think about what would happen to the other planets and celestial bodies if the earth were the size of a marble. He finds out that the asteroid Ceres would only be \begin{align*}2.9 \times 10^{-2}\end{align*}. Here is his question.

If the asteroid Ceres were \begin{align*}2.9 \times 10^{-2}\end{align*}, what size would that be as a decimal?

Aron is puzzled, he knows that he will need to multiply something, but isn't sure how to do it.

In this Concept, you will learn all about how to multiply by decimal powers of ten. Then you will be able to understand Aron's dilemma and how to solve it.

### Guidance

Previously we worked on how to use mental math to multiply by whole number powers of ten. Well, you will find that we can do the same thing with decimal powers of ten.

How does this happen when you multiply a decimal by a decimal power of ten?What changes?

When multiplying by a power of ten, we moved the decimal point to the right the same number of zeros as there was in the power of ten.

\begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} move the decimal to the right two places.

When we have what appears to be a power of ten after a decimal point, we we only move the decimal one place to the left.

Why?

.10, .100, .1000 appear to all be powers of ten, but they are actually all the same number. We can keep adding zeros in a decimal, but they still are all the same. They all equal .10.

Therefore, if you see a .1 with zeros after it, you still move the decimal point one place to the left, no matter how many zeros there are.

\begin{align*}.10 \times 4.5 & = .45 \\ .100 \times 4.5 & = .45\end{align*}

Try a few on your own. Find each product.

#### Example A

.10 \begin{align*}\times\end{align*} 6.7 \begin{align*}=\end{align*} _____

Solution: .67

#### Example B

.100 \begin{align*}\times\end{align*} .45 \begin{align*}=\end{align*} _____

Solution: .045

#### Example C

.10 \begin{align*}\times\end{align*} 213.5 \begin{align*}=\end{align*} _____

Solution: 21.35

Now let's go back to Aron. His problem involves scientific notation. Scientific notation is a something that you will learn about in another Concept, but for now, you can use what you have just learned about multiplying by decimals powers of ten to help Aron solve his dilemma. Here is his question once again.

If the asteroid Ceres were \begin{align*}2.9 \times 10^{-2}\end{align*}, what size would that be as a decimal?

Remember that the negative 2 exponent tells us how many places to move the decimal point to the left.

\begin{align*}2.9 \times 10^{-2} = .029\end{align*}

This is the solution.

### 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.

.100 \begin{align*}\times\end{align*} 6.734 \begin{align*}=\end{align*} _____

To solve this problem, we have to move the decimal point two places to the left. We do this because we are multiplying 6.734 times a decimal power of ten.

The answer is \begin{align*}.6734\end{align*}.

### Practice

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

1. 3.2 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

2. .678 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

3. 2.123 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

4. .890 \begin{align*}\times\end{align*} .1000 \begin{align*}=\end{align*} ______

5. 5 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

6. 7.7 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

7. 12 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

8. 456.8 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

9. .8 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

10. 4.56 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

11. 8.678 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

12. 16.608 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

13. 22.689 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______

14. 13.678 \begin{align*}\times\end{align*} .1000 \begin{align*}=\end{align*} ______

15. 45 \begin{align*}\times\end{align*} .1000 \begin{align*}=\end{align*} ______

16. 891 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______

### Vocabulary Language: English

Powers of ten

Powers of ten

The powers of ten are 10, 100, 1000, 10,000, etc. They are ten to the first power, ten to the second power, ten to the third power, etc.
Scientific Notation

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.