5.5 Multiplying and Dividing by Decimal Powers of Ten

Learning Objectives

Use powers of ten to multiple.
Use scientific notation to write long numbers in a short form.

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.

The diameter of the earth is about 13,000 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 10 times as large?
What would the diameter of the earth be if it were 1,000 times larger?

Kailey is puzzled and stops to think about her answer.

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 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!

Guided Learning

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 A

$3.4 \times 10 & = 34\\3.45 \times 100 & = 345\\.367 \times 10 & = 3.67\\.45 \times 1000 & = 450$

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.

Guided Practice

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

.23 $\times$ 10 _____
34.567 $\times$ 100 _____
127.3 $\times$ 10 _____

Now take a minute to check your work with a friend.

Solutions:

2.3
3,456.7
1,273

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.

$\times$ 100 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 B

$.10 \times 4.5 & = .45 \\.100 \times 4.5 & = .45$

Guided Practice

Try a few on your own.

.10 $\times$ 6.7 _____
.100 $\times$ .45 _____
.10 $\times$ 213.5 _____

Check your work. Did you complete these problems using mental math?

Solutions:

.67
.045
21.35

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 C

$2.5 \div 10 & = .25\\2.5 \div 100 & = .025\\2.5 \div 1000 & = .0025$

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.

Guided Practice

Use mental math to divide the following decimals.

4.5 $\div$ 10 _____
.678 $\div$ 1000 _____
87.4 $\div$ 100 _____

Double check your work with a friend. Were you able to mentally divide by a power of ten?

Solutions:

45
678
8,740

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.

$5.2 \div .10 & = 52\\5.2 \div .100 & = 52\\5.2 \div .1000 & = 52$

Once you have learned the rule, you can use mental math to complete the division of decimals by a power of ten.

Guided Practice

Practice using mental math to divide these decimals.

.67 $\div$ .10 _____
12.3 $\div$ .100 _____
4.567 $\div$ .1000 _____

Stop and check your work.

Solutions:

.067
1.23
.4567

Write in Scientific Notation

What is scientific notation?

Scientific Notation is a shortcut for writing longer 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.

Let’s look at an example.

Example D

4,500 = 4.5 x 102

This example has a whole number and not a decimal. We start with a number called 4,500, this has two decimal places in it. Therefore, we are going to say that if we multiplied 4.5 by 10 squared, we would have 4,500 as our number.

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

Guided Practice

Practice writing a few of these decimals in scientific notation.

2,200,000,000 = _____
3,800,000 = _____
34,000,000,000 = _____

Take a few minutes to check your work.

Solutions:

2,200,000, 000 = _____ 2.2 x 109
3,800,000 = _____ 3.8 x 106
34,000,000,000 = _____ 3.4 x 1010

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 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 13,000 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 10 times as large?
What would the diameter of the earth be if it were 1,000 times larger?

Kailey is puzzled and stops to think about her answer.

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.

13,000 $\times$ 10 130,000 km

In scientific notation it would be 1.3 x 105

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 larger. To complete this problem, Kailey needs to multiply the diameter of the earth by 100. She will move the decimal point two places to the right.

13,000 x 1,000 13,000,000

In scientific notation it would be 1.3 x 107

Review

Product is the answer to a multiplication problem.
Quotient is the answer to a division problem.
Exponential notation is an easier way to write a number as a product of many factors.

Scientific Notation

A shortcut for writing longer numbers and decimals.

Powers of Ten

When a product can only be derived by multiplying by 10. Examples 10, 100, 1,000, etc.

Video Resources

Dividing by Powers of Ten Video

Multiplying by Powers of Ten Video

Learning Objectives

Introduction

Guided Learning

Use Mental Math to Multiply Decimals by Whole Number Powers of Ten

Example A

Guided Practice

1,273

Use Mental Math to Multiply Decimals by Decimal Powers of Ten

Example B

Guided Practice

Use Mental Math to Divide Whole Numbers by Whole Number Powers of Ten

Example C

Guided Practice

8,740

Use Mental Math to Divide Whole Numbers by Decimal Powers of Ten

Guided Practice

.4567

Write in Scientific Notation

Example D

Guided Practice

Real Life Example Completed

Review

Video Resources