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