Multiplying Decimals by Powers of 10 — Rules & Examples

Multiplying and Dividing Decimals by Powers of $10$

When you multiply a decimal by a power of $10,$ simply move the decimal place to the right as many places as there are $0\text{s}$ in the power of $10.$

When you divide a decimal by a power of $10,$ simply move the decimal place to the left as many places as there are $0\text{s}$ in the power of $10.$

Multiplying Decimals by $10$

Multiplying Decimals by $100$

Dividing Decimals by $10$

Dividing Decimals by $100$

(Video) Multiplying and Dividing Decimals by $10$, $100$, and $1{,}000$

by Richard Russell

Here's a great video in which Richard shows you how to multiply and divide decimals by powers of $10$. He uses a visual method to help you understand what is happening to numbers as you do this.

$\large{\text{Summary}}$

One of the important parts of the decimal system is learning when to move the decimal point.

When we multiply a decimal by a power of $10$, we're making the number larger. When we divide by a multiple of $10$, we're making the number smaller.

Multiplying

The first example he looks at is $3.6\times10$. We want to put this number into the following grid.

Here, the decimal place will stay put. Putting in $3.6$, we get:

Now, we want to multiply by $10$. Multiplying will make our decimal larger, meaning we want to move the digits to the left. We mean the digits move to the left of the decimal point. How far to the left though? You move it the number of spaces equal to the number of zeros in the power of $10$. Since $10$ has one zero, we move one space to the left.

We now have our solution! $$3.6\times10=36$$

Dividing

Dividing is the same, we just move the digits in the opposite direction. The example he looks at is: $4.63\div100$. Start by placing your decimal number in the grid.

Now, we divide by $100$. Since it's division, we move the digits to the right. And, because $100$ has two zeros, we move each digit two spaces to the right.

Since there is empty space between the digits and the decimal, we fill it in with $0$s.

We have our solution! $$4.63\div100=0.0463$$

If you're familiar with scientific notation, we could also write this as

$$4.63 \times 10^{-2}$$

Think of the $-2$ as a signal. The decimal was moved two places to the left!

Although this explanation focuses on $10$ and $100$, we can do larger powers of $10$ too. For instance, multiplying

$$4.63 \times 10^4$$

puts the $4$ in the ten thousands place. It also puts the $6$ in the thousands place. Finally, the $3$ goes in the hundreds place. Since we have some gaps we just fill them with zeros. The result is

$$4.63 \times 10^4 = 46{,}300$$

This result is a whole number. That means we don't write any digits after the decimal point. In fact, we don't even have to write the decimal point! It's invisible, right after the last zero.

Multiplying or Dividing Decimals by $10$ or $100$

If you multiply a decimal by $10$, the decimal point will move one place to the right.

If you divide a decimal by $10$, the decimal point will move one place to the left.

If you multiply or divide by any power of $10$, it will move a number of spaces equal to the number of $0$’s.

Example 1

What is $15.678 \times 100$?

We must first note that $15.678 \times 100$ is multiplication, so the decimal place for $15.678$ will move to the right.

Next, the multiple of ten is $100$, which has two zeros. Therefore, we move the decimal place in $15.678$ to the right by two places. $$15.678 \rightarrow 1567.8$$

Example 2

What is $109.42 \div 100$?

We must first note that $109.42 \div 100$ is division, so the decimal place for $109.42$ will move to the left.

Next, the multiple of ten is $100$, which has two zeros. Therefore, we move the decimal place in $109.42$ to the left by two places. $$109.42 \rightarrow 1.0942$$

We are now going to take a look at some of the types of problems you might encounter about multiplying or dividing decimals by $10$ or $100$.

When you divide a decimal by $10$ the decimal point will move one place to the left.

We must first note that $831 \div 10{,}000$ is division, so the decimal place for $831$ will move to the left.

Next, the multiple of ten is $10{,}000$, which has four zeros. Therefore, we move the decimal place in $831$ to the left by $4$ places. $$831 \rightarrow 0.0831$$

When you multiply a decimal by $10$, the decimal point will move one place to the right.

We must first note that $0.371 \times 1{,}000$ is multiplication, so the decimal place for $0.371$ will move to the right.

Next, the multiple of ten is $1{,}000$, which has three zeros. Therefore, we move the decimal place in $0.371$ to the right by $3$ places. $$0.371 \rightarrow 371$$

When you multiply a decimal by $10$, the decimal point will move one place to the right.

When you divide a decimal by $10$ the decimal point will move one place to the left.

This problem uses both multiplication and division.

First, note that $165 \times 100$ is multiplication, so the decimal place for $165$ will move to the right.

Next, the multiple of ten is $100$, which has two zeros. Therefore, we move the decimal place in $165$ to the right by $2$ places. $$165 \rightarrow 16{,}500$$

Next, we have to solve for $16{,}500 \div 1{,}000$.

We note that $16{,}500 \div 1{,}000$ is division, so the decimal place for $16{,}500$ will move to the left.

Next, the multiple of ten is $1{,}000$, which has three zeros. Therefore, we move the decimal place in $16500$ to the left by $3$ places. $$16{,}500 \rightarrow 16.5$$