# Finding a Power of a Power

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 a Power of a Power

Finding a power of a power is easy.

#### An Example Question

Use the law of exponents to find the power of a power below.

Step 1

Find the exponents. In our example, the exponents are 2 and 3.

Step 2

Multiply the exponents together.

2 × 3 = 6
Step 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 a 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 bracket 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.

# A Real Example of How to Find a Power of a Power

The slider below shows another real example of how to find a power of a power.

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