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Multiplication and Division by Powers of 10
> ... Math > Multiplication and Division > Multiplication and Division by Powers of 10

Introduction

In this lesson, you will learn about multiplication and division by powers of 10. A power of 10 is a number that can be written as 10 raised to a power or exponent. Multiplying or dividing by these powers simply requires you to move the decimal place of the number that you’re multiplying or dividing.

This video illustrates the lesson material below. Watching the video is optional.

Powers of 10

A power of 10 is a number that can be written as 10 raised to a power or an exponent. For example:

\(10^{1} = 10\)

\(10^{2} = 10\times 10 = 100\)

\(10^{3}=10\times10\times10=1000\)

One thing to notice is that the number of the exponent, or the number of powers, is the same as the number of zeros after the one in the answer. \(10^2\) has two zeros, \(10^3\) has three zeros, etc. The concept of multiplying by 10, or the power of 10, is that you take the number of the power and put the same number of zeros after it.

Example 1
\(2\times10=20\)

The result is the 2 followed by a zero, because 10 has only one zero.

Multiplying Decimal Points

Referencing the example above, now consider what this means for a decimal point.

Example 2
In the following equation, 10 represents the same thing as \(10^1\):

\begin{align*}
1.0\times10=10.0\\
\end{align*}

The decimal point was moved over one space to the right, from 1.0 to 10.0.

Example 3
\begin{align*}
1.0\times100=100.0
\end{align*}

This time, the 100 represents \(10^2\). Because there are two zeros after the one, the decimal would move over two spaces to the right.

Example 4
\begin{align*}
2\times10=20
\end{align*}

In reality, the equation is multiplying \(2.0\times10\). When you multiplied 2.0, you moved the decimal one space to the right, making the answer 20. Any time you multiply by a power of 10, move the decimal point one space to the right for every zero that follows the 1.

Example 5
The same concept applies when you change the decimal point.

\begin{align*}
3.8\times10=38.0
\end{align*}

Because you are multiplying by 10, or \(10^{1}\), move the decimal point one space to the right, making the answer 38.

Example 6
This example follows the same rule.

\begin{align*}
453.75\times100=45375.0
\end{align*}

Because you are multiplying by 100 or \(10^{2}\), move the decimal point two spaces to the right.

Division

This rule is opposite for division. While multiplication moves the decimal point one space to the right, division moves the decimal point one space to the left.

Example 7
If you have $100 dollars and you divide it into either 10 or 100 equal segments, you will have the following:

\begin{align*}
100\div10&=10.0\\\\
100\div100 &=1.00
\end{align*}

Example 8
This example shows how this works with more complex numbers. Following the same pattern used before, simply move the decimal point one space to the left if divided by 10 or two decimal places to the left if divided by 100:

\begin{align*}
527.38\div10&=52.738\\\\
438\div100 &=4.38
\end{align*}

Things to Remember


  • When multiplying by 10, or looking at the power of 10, take the number of the power and put the same number of zeros after the one.
  • When multiplying, move the decimal point to the right.
  • When dividing, move the decimal point to the left.

Practice Problems

Evaluate the following expressions:
  1. \(86 × 10 = ?\) (
    Solution
    x
    Solution: 860
    Details:
    In this image we see the numbers 86 and 860 written inside a table. The columns of the table are labeled with the number place values. The number 86 is in the first row of the table. The 8 is in the tens column and the 6 is in the ones column. In the next row down is the number 860. The 8 is in the hundreds column while the 6 is in the tens column and the 0 is in the ones column. 
    The place value chart shows how the original number, 86, when multiplied by 10 moves from 8 tens and 6 ones to 8 hundreds, 6 tens, and 0 ones. When multiplying by 10, each number moves one space to the left in the place value chart.
    )
  2. \(295 ÷ 10 = ?\) (
    Solution
    x
    Solution: 29.5
    Details:
    This image shows the numbers 295 and 29 point 5 written inside a table. The columns of the table are labeled with the number place values. The number 295 is in the first row of the table. The 2 is in the hundreds column, the 9 is in the tens column and the 5 is in the ones column. In the next row down is the number 29 point 5. The 2 is in the tens column, the 9 is in the ones column. There is a decimal point to the right of the 9, and then the 5 is to the right of that in the tenths column. 
    As shown on the place value chart, when 295 is divided by 10, 2 hundreds, 9 tens, and 5 ones become 2 tens, 9 ones, and 5 tenths. Dividing by 10 moved each number one place to the right.
    )
  3. \(9.72 × (10) = ?\) (
    Solution
    x
    Solution:
    97.2
    )
  4. \(54.6 ÷ 10 = ?\) (
    Video Solution
    x
    Solution: 5.46
    Details:

     | Transcript)
  5. \(3.95 × 100 = ?\) (
    Video Solution
    x
    Solution: 395
    Details:

     | Transcript)
  6. \(17 ÷ 100 = ?\) (
    Solution
    x
    Solution:
    0.17
    )
    Division
    Multiplication with Decimal Numbers