4.5: Multiplying and Dividing by Decimal Powers of Ten
Introduction
The Earth’s Diameter
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 is asked specific questions. Here they are:
- What would the diameter of the earth be if it were ten times as large?
- What would the diameter of the earth be if it were 100 times smaller?
Kailey is puzzled and stops to think about her answer.
Meantime, Aron is curious about what Kailey is working on. He comes over next to her and begins working on a different activity. 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.
1. If the asteroid Ceres were \begin{align*}2.9 \times 10^{-2}\end{align*}, what size would that be as a decimal?
Aron looks at Kailey with a blank stare.
They are both stuck!
This is where you come in. Kailey will need to know how to multiply and divide by multiples of ten to complete her activity. Aron will need to remember how to work with scientific notation to complete his activity.
Pay close attention in this lesson and you will be able to help them by the end!
What You Will Learn
In this lesson you will learn how to complete the following:
- Use mental math to multiply decimals by whole number powers of ten.
- Use mental math to multiply decimals by decimal powers of ten.
- Use mental math to divide decimals by whole number powers of ten.
- Use mental math to divide decimals by decimal powers of ten.
- Write in scientific notation.
Teaching Time
I. Use Mental Math to Multiply Decimals by Whole Number Powers of Ten
This lesson 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?
To understand this, let’s look at a few examples.
Example
\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.
- .23 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} _____
- 34.567 \begin{align*}\times\end{align*} 100 \begin{align*}=\end{align*} _____
- 127.3 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} _____
Now take a minute to check your work with a friend.
II. Use Mental Math to Multiply Decimals by Decimal Powers of Ten
How does this change when you multiply a decimal by a decimal power of ten? 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? Let’s look at an example to understand 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.
Example
\begin{align*}.10 \times 4.5 & = .45 \\ .100 \times 4.5 & = .45\end{align*}
Try a few on your own.
- .10 \begin{align*}\times\end{align*} 6.7 \begin{align*}=\end{align*} _____
- .100 \begin{align*}\times\end{align*} .45 \begin{align*}=\end{align*} _____
- .10 \begin{align*}\times\end{align*} 213.5 \begin{align*}=\end{align*} _____
Check your work. Did you complete these problems using mental math?
III. Use Mental Math to Divide Whole Numbers by Whole Number Powers of Ten
You just finished using mental math when multiplying, you can use mental math to divide by whole number powers of ten too.
Here are a few examples of 2.5 divided by whole number powers of ten. See if you can see the pattern.
Example
\begin{align*}2.5 \div 10 & = .25\\ 2.5 \div 100 & = .025\\ 2.5 \div 1000 & = .0025\end{align*}
What is the pattern?
When you divide by a power of ten, you move the decimal point to the left according to the number of zeros that are in the power of ten that you are dividing by.
Once you have learned and memorized this rule, you will be able to divide using mental math.
Notice that division is the opposite of multiplication. When we multiplied by a power of ten we moved the decimal point to the right. When we divide by a power of ten, we move the decimal point to the left.
Use mental math to divide the following decimals.
- 4.5 \begin{align*}\div\end{align*} 10 \begin{align*}=\end{align*} _____
- .678 \begin{align*}\div\end{align*} 1000 \begin{align*}=\end{align*} _____
- 87.4 \begin{align*}\div\end{align*} 100 \begin{align*}=\end{align*} _____
Double check your work with a friend. Were you able to mentally divide by a power of ten?
IV. Use Mental Math to Divide Whole Numbers by Decimal Powers of Ten
You have already learned how to multiply by what appears to be a power of ten after a decimal place. Remember that all powers of ten that you see written to the right of a decimal point are equal.
.10 = .100 = .1000 = .10000
When we multiply by this power of ten to the right a decimal point, we move the decimal point one place to the left. When we divide by a power of ten to the right a decimal point, we are going to move the decimal point one place to the right. If you think about this it makes perfect sense. The powers of ten written to the right of a decimal point are all equal. It doesn’t matter if you are multiplying or dividing by .10 or .100 or .1000. Division is the opposite of multiplication so you move the decimal point one place to the right.
\begin{align*}5.2 \div .10 & = 52\\ 5.2 \div .100 & = 52\\ 5.2 \div .1000 & = 52\end{align*}
Once you have learned the rule, you can use mental math to complete the division of decimals by a power of ten.
Practice using mental math to divide these decimals.
- .67 \begin{align*}\div\end{align*} .10 \begin{align*}=\end{align*} _____
- 12.3 \begin{align*}\div\end{align*} .100 \begin{align*}=\end{align*} _____
- 4.567 \begin{align*}\div\end{align*} .1000 \begin{align*}=\end{align*} _____
Stop and check your work.
V. Write in Scientific Notation
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 a few of these numbers in scientific notation.
- .0012 = _____
- 78,000,000 = _____
- .0000023 = _____
Take a few minutes to check your work.
Real Life Example Completed
The Earth’s Diameter
You have finished learning about division by powers of ten. Astronomers use scientific notation, multiplication and division by powers of ten all the time. Think about it, they work with very large and very small decimals.
Now you are ready to help Kailey and Aron with their work. Here is the problem once again.
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 is asked specific questions. Here they are:
- What would the diameter of the earth be if it were ten times as large?
- What would the diameter of the earth be if it were 100 times smaller?
Kailey is puzzled and stops to think about her answer.
Meantime, Aron is curious about what Kailey is working on. He comes over next to her and begins working on a different activity. 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 looks at Kailey with a blank stare.
They are both stuck!
First, let’s take a minute to underline the important information.
Let’s 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!
Kailey’s second question asks if what the diameter of the earth would be if it were 100 times smaller. To complete this problem, Kailey needs to divide the diameter of the earth by 100. She will move the decimal point two places to the left.
12,756.3 \begin{align*}\div\end{align*} 100 \begin{align*}=\end{align*} 127.563
Wow! The earth went from being in the ten-thousands to being in the hundreds. Think about how much smaller that is!
Let’s not forget about Aron. His problem involves scientific notation. 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*}
Aron is excited to understand scientific notation. Here is another fact that he discovers at his work station.
If a Neutron Star was \begin{align*}6.17 \times 10^{-4}\end{align*} inches that would mean that it was .000617 inches. That is a very small star!!!
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.
Resources
If you found the information on Astronomy useful, you can go to the following websites for more information.
- www.wikianswers.com – this site will answer any question that you may have about the solar system.
- www.janus.astro.umd.edu/AW/awtools – this is a website for the Astronomy Workshop which has great interactive activities using mathematics and astronomy.
Technology Integration
James Sousa Dividing by Powers of Ten
Other Videos:
http://www.mathplayground.com/howto_dividedecimalspower10.html – Good basic video on how to divide decimals by a power of ten
Time to 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*} ______
Directions: Use mental math to multiply each decimal by a decimal power of ten.
8. 3.2 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______
9. .678 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______
10. 2.123 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______
11. .890 \begin{align*}\times\end{align*} .1000 \begin{align*}=\end{align*} ______
12. 5 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______
13. 7.7 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______
14. 12 \begin{align*}\times\end{align*} .10 \begin{align*}=\end{align*} ______
15. 456.8 \begin{align*}\times\end{align*} .100 \begin{align*}=\end{align*} ______
Directions: Use mental math to divide each decimal by a power of ten.
16. 3.4 \begin{align*}\div\end{align*} 10 \begin{align*}=\end{align*} ______
17. 67.89 \begin{align*}\div\end{align*} 100 \begin{align*}=\end{align*} ______
18. 32.10 \begin{align*}\div\end{align*} 10 \begin{align*}=\end{align*} ______
19. .567 \begin{align*}\div\end{align*} 100 \begin{align*}=\end{align*} ______
20. .87 \begin{align*}\div\end{align*} 1000 \begin{align*}=\end{align*} ______
Directions: Use mental math to divide each decimal by a decimal power of ten.
21. 6.7 \begin{align*}\div\end{align*} .10 \begin{align*}=\end{align*} ______
22. .654 \begin{align*}\div\end{align*} .100 \begin{align*}=\end{align*} ______
23. 2.1 \begin{align*}\div\end{align*} .10 \begin{align*}=\end{align*} ______
24. 4.32 \begin{align*}\div\end{align*} .1000 \begin{align*}=\end{align*} ______
25. .98765 \begin{align*}\div\end{align*} .10 \begin{align*}=\end{align*} ______
Directions: Write each decimal in scientific notation.
26. .00056
27. 98,000
28. .024
29. 2,340,000,000
30. .00000045
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