﻿ Multiplying and Dividing Decimals by Powers of Ten
MULTIPLYING AND DIVIDING DECIMALS BY POWERS OF TEN

In any multiplication problem, the numbers being multiplied are called the factors.
For example, in the multiplication problem $\,23.7\times 10\,$, the factors are $\,23.7\,$ and $\,10\,$.

To multiply a decimal by powers of ten, you move the decimal point one place to the right for each factor of ten.

Recall that $\,10^5\,$ is a shorthand for $\,5\,$ factors of $\,10\,$: $\,10^5 = 10\cdot 10\cdot 10\cdot 10\cdot 10\,$.
Similarly, $\,10^n\,$ is a shorthand for $\,n\,$ factors of $\,10\,$.

The $\,\times\,$ symbol is used for multiplication in these problems, because the centered dot is too easily confused with the decimal point.

EXAMPLES:
$23.19 \times 10 = 231.9$
Move the decimal point one place to the right.
$7.001 \times 10^3 = 7001$
Move the decimal point three places to the right.
$0.03 \times 10^4 = 300$
Move the decimal point four places to the right, inserting zeros as needed.

To divide a decimal by powers of ten, you move the decimal point one place to the left for each factor of ten.

In this web exercise, division is denoted using either the ‘ $\,\div\,$’ symbol, or a horizontal fraction bar.

EXAMPLES:
$23.1 \div 10 = 2.31$
Move the decimal point one place to the left.
$\displaystyle\frac{7.001}{10^3} = 0.007001$
Move the decimal point three places to the left, inserting zeros as needed.

Make sure you understand why this works!
For example, when $\,2.37\,$ is divided by $\,10\,$, the $\,2\,$ ones should turn into $\,2\,$ tenths.
Moving the decimal point one place to the left accomplishes this.

Master the ideas from this section

When you're done practicing, move on to:
Changing Decimals to Percents

Here, you will practice multiplying and dividing decimals by powers of ten.
That is, type the answer to   $\,631.47\times 10^3\,$   as   $\,631470\,$,   not   $\,631{,}470\,$.