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Powers of a Power
(KS3, Year 7)

homesitemaparithmeticfinding 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 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. 2 squared cubed

Step-by-Step:

1

Find the exponents. In our example, the exponents are 2 and 3. 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. 2 to the 6

Answer:

We have found the power of the 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 brackets 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.

Lesson Slides

The slider below shows another real example of how to find a power of a power.
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This page was written by Stephen Clarke.

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