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Powers of a Power
(KS3, Year 7)
The Lesson
We can find a power of a power. In this case, a power (with an exponent) is itself raised to an exponent. To find a power of a power, multiply the exponents together.This is a law of exponents.
How to Find the Power of a Power
Finding a power of a power is easy.Question
Use the law of exponents to find the power of the power below.StepbyStep:
1
Find the exponents. In our example, the exponents are 2 and 3.
2
Multiply the exponents together.
2 × 3 = 6
3
Make the answer from Step 2 (6) the exponent of the base that has been raised to the exponent.
Answer:
We have found the power of the power.Understanding Powers of a Power
Let us look at the rule for finding a power of a power, using the example above:Firstly, let us look at what is to the left of the equals sign (=):

Inside the brackets is a power, 2^{2}. It consists of a base (2) raised to an exponent (2).
2^{2} means 2 is multiplied by itself 2 times:2^{2} = 2 × 2 
This power becomes the base of another power, (2^{2})^{3}. Here, the base is 2^{2} and the exponent is 3.
(2^{2})^{3} means 2^{2} is multiplied by itself 3 times:(2^{2})^{3} = 2^{2} × 2^{2} × 2^{2}By writing out this in full, we see that the left hand side is equal to 2 multiplied by itself 6 times: 2^{6}.2^{2} × 2^{2} × 2^{2} = (2 × 2) × (2 × 2) × (2 × 2) 2^{2} × 2^{2} × 2^{2} = 2 × 2 × 2 × 2 × 2 × 2 2^{2} × 2^{2} × 2^{2} = 2^{6}
 2^{2 × 3} is a power. It consists of a base (2) raised to an exponent (2 × 3). Clearly, 2^{2 × 3} is equal to 2^{6}. The left hand side of the equation equals the right hand side.
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