Multiplying by Powers of 10

Need some basic practice with place value first? Identifying Place Values

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, $\,237 \cdot 10^7 = 2{,}370{,}000{,}000\,.$ (Count the seven zeros after the ‘$\,237\,$’!)

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. For more information, see Scientific Notation.

Practice