Follow Us on Twitter

How to Find a Power of a Power (Mathematics Lesson)

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.

2 to the m all to the n is equal to 2 to the m times n

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.

2 squared cubed
Step 1

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

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.

2 to the 6

We have found the power of a power.

2 squared cubed equals 2 to the 6

Understanding Powers of a Power

Let us look at the rule for finding a power of a power, using the example above:

bases and exponents

Firstly, let us look at what is to the left of the equals sign (=):

2 to the 2 to the 3
  • Inside the bracket is a power, 22. It consists of a base (2) raised to an exponent (2).

    2 squared powers, bases and exponents

    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.

    2 squared cubed powers, bases and exponents

    (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 (=):

2 to the 2 times 3
  • 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.

Interactive Test
  show
 

Here's a second test on finding powers of a power.
Here's a third test on finding powers of a power.