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# Multiplying and Dividing Exponents. What are the rules for multiplying and dividing powers?

## Multiplication And Division Power Rules

The quick rules for multiplying and dividing exponent that have the same base numbers are as follows:

**a ^{m}×a^{n} = a^{m+n}** (so you just need to add the exponents)

**a ^{m} ÷a^{n} = a^{m-n}**(so take the second exponent from the first)

So basically, add the exponents for multiplying and work out the difference for dividing.

**Example 1**

Simplify x⁷ × x^{⁴}

Since you are multiplying add the exponents:

x⁷ × x^{⁴} = x^{7+4} = x^{11}

**Example 2**

Simplify y^{8} × y

Again you are multiplying the exponents so add them together. Also y has an exponent of 1:

y^{8} × y = y^{8+1} = y^{9}

**Example 3**

Simplify x^{12} ÷ x^{9}

This time you are dividing the terms so work out the difference:

x^{12} ÷ x^{9} = x^{12-9} = x^{3}

**Example 4**

Simplify a^{5}/a^{7}

This time you are dividing the terms so work out the difference again:

a^{5}/a^{7}= a^{5-7} = a^{-2}

**Example 5**

Simplify x^{6}y^{2} × x^{4}y^{10}.

Here, you need to add the exponents of the x terms and add the exponents of the y terms:

x^{6} × x^{⁴} = x^{10}

y^{2} × y^{10} = y^{12}

So if you put these together you get a final answer of:

x^{10} y^{12}

**Example 6**

Simplify x^{8}y^{7} ÷ x^{4}y^{3}.

Here, you need to take the exponents of the x terms and also take the exponents of the y terms as you are dividing:

x^{8} ÷ x^{4} = x^{8-4} = x^{4}

Y^{7} ÷ y^{3} = y^{4}

So if you put these together you get a final answer of:

x^{4} y^{4}

**Example 7**

Simplify 7c^{6}d^{8}e^{4} × 8c^{3}de^{-9}

This is not as hard as it looks.

First multiply 7 by 8 to give 56 (don’t add these together as they are not exponents). This will go at the start of your answer.

Next work out the exponents of c, d and e by adding the exponents as it’s a multiplication:

c^{6}×c^{3} = c^{9}

d^{8}×d^{1} = d^{9}

e^{4}×e^{-9} = e^{-5}

So if you put all of these answers together you get the expression:

56c^{9}d^{9}e^{-5}

So to summarise, multiplying exponents you add the powers and dividing exponents take the powers.

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