Multiplying and dividing by powers of 10

​​In a nutshell

Powers of $$10$$​ refer to $$10, 100, 1000$$​ and so on. To multiply and divide by powers of $$10$$​, you have to use place values to move the decimal place a certain number of steps to the left or right.

Multiplying by powers of 10

To multiply by powers of $$10$$​, follow this procedure:

procedure

1. ​Count how many zeros there are in the power of $$10$$​.
   
2. Move the decimal point to the right by the same amount.

Note: If the number is a whole number, then add "$$.000...$$", adding as many zeros as required.

Example 1

What is $$3.2\times1000$$?

First, count how many zeros are in $$1000$$. Then, write $$3.2$$ as $$3.2000..$$ and move the decimal point $$\textbf{3}$$ places to the right. Finally, get rid of the extra zeros past the decimal point.

​​

Dividing by powers of 10

To divide by powers of $$10$$​, it is exactly the same as multiplying, except the decimal point moves the other way.

procedure

1. ​Count how many zeros there are in the power of 10.
   
2. Move the decimal point to the left by the same amount, adding zeroes to the left if necessary.

​

Example 2

What is $$25\div100000$$?

First, count how many zeros there are in $$100000$$. Then, write $$25$$​ as $$...000025.0$$, and move the decimal point 5 places to the left. Get rid of the unnecessary zeros.​​

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Multiplying and dividing by multiples of powers of 10

Multiples of powers of $$10$$​ refers to numbers such as: $$30000, 20, 800$$​ and so on. To multiply and divide by these numbers, first you have to multiply or divide by the multiple (the number "attached" to the power of $$10$$​).

Example 3

What is $$400\div50000$$​?

First, write $$50000$$ as $$5\times10000$$.

To compute $$400\div50000$$, first do $$400\div5$$ and then divide the answer by $$10000$$, using the method above:

​$$400\div5=80$$​​

​$$80\div10000=0.\overset{\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}{0\thinspace 0\thinspace 8\thinspace 0}$$​​

​$$80\div10000=0.008$$​​

​$$\underline{400\div50000=0.008}$$​​