# 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. # 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. 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, 22. It consists of a base (2) raised to an exponent (2). 22 means 2 is multiplied by itself 2 times:
22 = 2 × 2
• This power becomes the base of another power, (22)3. Here, the base is 22 and the exponent is 3. (22)3 means 22 is multiplied by itself 3 times:
(22)3 = 22 × 22 × 22
By writing out this in full, we see that the left hand side is equal to 2 multiplied by itself 6 times: 26.
22 × 22 × 22 = (2 × 2) × (2 × 2) × (2 × 2) 22 × 22 × 22 = 2 × 2 × 2 × 2 × 2 × 2 22 × 22 × 22 = 26
Let us look at the right hand side of the equals sign (=): • 22 × 3 is a power. It consists of a base (2) raised to an exponent (2 × 3). Clearly, 22 × 3 is equal to 26. The left hand side of the equation equals the right hand side.

## Lesson Slides

The slider below shows another real example of how to find a power of a power. Open the slider in a new tab
Help Us To Improve Mathematics Monster