MULTIPLYING BY POWERS OF TEN

In the base ten number system, it is extremely easy to multiply by powers of ten.
To multiply by   $\,10^1 = 10\,$,   put $\,1\,$ zero at the end of the number:    $237\cdot 10 = 2{,}370\,$
To multiply by $\,10^2 = 100\,$, put $\,2\,$ zeros at the end of the number:   $237\cdot 100 = 23{,}700\,$

To multiply by $\,10^n\,$ (which is $\,1\,$ followed by $\,n\,$ zeroes),
put $\,n\,$ zeros at the end of the number.
For example, $\,237 \cdot 10^7 = 2{,}370{,}000{,}000\,$.

Think about why this is so easy!
When, say, $\,237\,$ is multiplied by $\,10\,$:

• the $\,2\,$ hundreds become $\,2\,$ thousands;
• the $\,3\,$ tens become $\,3\,$ hundreds;
• the $\,7\,$ ones become $\,7\,$ tens.
Each digit needs to shift into the next-left place value.
Putting the zero at the end of the number accomplishes this.

In this exercise, multiplication is denoted in two ways:

• using a centered dot:   $\,237\cdot 10^3\,$
• using the ‘times’ symbol:   $\,237\times 10^3\,$
This context—multiplying by powers of ten—is one of the only places in algebra and beyond
where use of the ‘$\,\times\,$’ symbol for multiplication is appropriate.

Master the ideas from this section

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

Here, you will practice multiplying by powers of ten.
That is, type the answer to $\,631\times 10^3\,$ as $\,631000\,$, not $\,631{,}000\,$.