# Powers in Brackets: How to Use the Bracket Power Rule

Here you will be shown how to simplify expressions involving brackets and powers. The general rule is:

(x^{m})^{n }= x^{mn}

So basically all you need to do is multiply the powers. This may also be called the exponent bracket rule or indices bracket rule as powers, exponents and indices are all the same thing.

Let’s take a look at some examples involving brackets and powers:

Example 1

Simplify (x^{5})^{4}.

So all you need to do is follow the rule given above by multiplying the powers together:

(x^{m})^{n }= x^{mn}

(x^{5})^{4 }= x^{5x4}=x^{20}

Example 2

Simplify (a^{7})^{3}

Again follow the bracket power rule by multiplying the powers:

(a^{7})^{3 }= a^{7x3}=a^{21}

The next example involvers a negative power, but the same rule can be applied.

Example 3

Simplify (y^{-4})^{6}

Again follow the bracket power rule by multiplying the powers:

(y^{-4})^{6 }= y^{-4x6}=y^{-24}

Remember that when you multiply a negative number by a positive number you get a negative answer.

On the next example there are two terms inside the bracket, but all you need to do is multiply both of the powers on the inside of the bracket by the power on the outside of the bracket. So you can change the above power rule to:

(x^{m}y^{n})^{p}= x^{mp}y^{np}

Example 4

Simplify (x^{6}y^{7})^{5}

Again follow the bracket power rule by multiplying the powers:

(x^{6}y^{7})^{5} = x^{6x5}y^{7x5} = x^{30}y^{35}

So all you need to do was multiply the 6 by 5 and the 7 by 5.

In the next two examples you will have a number in front of the algebra inside the bracket.

Example 5

Simplify (4x^{7})^{3}

Here you need to split this up as:

4^{3}(x^{7})^{3}

So the cube of 4 is 64 and (x^{7})^{3} can be simplified to x^{21}.

So the final answer you get is 64x^{21}.

If you didn’t like that method you could think that when you cube something you multiply it by itself three times. So (4x^{7})^{3} = 4x^{7}.4x^{7}.4x^{7}. And if you use the multiplication rule for powers and multiply the numbers together you get 64x^{21}.

Example 6

Simplify (9x^{8}y^{4})^{2}

Here you need to split this up as:

9^{2}(x^{8})^{2}(y^{4})^{2}

So the square of 9 is 81, (x^{8})^{2} can be simplified to x^{16} and (y^{4})^{2} = y^{8}

So the final answer you get is 81x^{16}y^{8}

Again, if you didn’t like the above method you could multiply 9x^{8}y^{4} by 9x^{8}y^{4} as when you square something it’s the same as multiplying the number by itself. You can then apply the multiplication power rule to simplify the algebra.

So to summarise the bracket power rule all you need to do is multiply the powers together.

## Questions & Answers

What should you do if the base and the index are not the same?

You should still be able to apply the bracket rule to this question as you just need to multiply the indices, the base number is not changed.

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