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Powers in Brackets: How to Use the Bracket Power Rule

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how-to-use-the-bracket-power-rule-example-simplify-x54-x20

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

(xm)n = xmn

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 (x5)4.

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

(xm)n = xmn

(x5)4 = x5x4=x20

Example 2

Simplify (a7)3

Again follow the bracket power rule by multiplying the powers:

(a7)3 = a7x3=a21

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:

(xmyn)p= xmpynp

Example 4

Simplify (x6y7)5

Again follow the bracket power rule by multiplying the powers:

(x6y7)5 = x6x5y7x5 = x30y35

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 (4x7)3

Here you need to split this up as:

43(x7)3

So the cube of 4 is 64 and (x7)3 can be simplified to x21.

So the final answer you get is 64x21.

If you didn’t like that method you could think that when you cube something you multiply it by itself three times. So (4x7)3 = 4x7.4x7.4x7. And if you use the multiplication rule for powers and multiply the numbers together you get 64x21.

Example 6

Simplify (9x8y4)2

Here you need to split this up as:

92(x8)2(y4)2

So the square of 9 is 81, (x8)2 can be simplified to x16 and (y4)2 = y8

So the final answer you get is 81x16y8

Again, if you didn’t like the above method you could multiply 9x8y4 by 9x8y4 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

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

Answer: 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.

Question: What if there is one base without indices in the bracket, such as (3x^4)^2?

Answer: First work out 3^2 = 9, and multiply the indices to give 8 (4 times 2).

So the final answer would be 9x^8.

Only multiply the indices together.

Question: What are the words in the BEDMAS anagram?

Answer: Brackets, Exponents, Division, Multiplication, Addition and Subtraction.

Question: What would (x-2) to power of 2 be?

Answer: This is a double bracket question (x-2)(x-2).

Expanding and simplifying will give x^2 -4x + 4.

Comments

Joan Whetzel on December 06, 2012:

Oh, you made this so easy to understand. Thanks.