Finding a Power of a Power in Algebra
(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.

a to the m all to the n is equal to a to the m times n This is a law of exponents.

How to Find the Power of a Power in Algebra

Finding a power of a power in algebra is easy.

Question

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

x to the 6

Answer:

We have found the power of the power.

x squared cubed equals x to the 6

Understanding Powers of a Power in Algebra

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 (=):

x to the 2 to the 3
  • Inside the brackets is a power, x2. It consists of a base (x) raised to an exponent (2).

    x squared powers, bases and exponents x2 means x is multiplied by itself 2 times:
    x2 = x × x
  • This power becomes the base of another power, (x2)3. Here, the base is x2 and the exponent is 3.

    x squared cubed powers, bases and exponents (x2)3 means x2 is multiplied by itself 3 times:
    (x2)3 = x2 × x2 × x2
    By writing out this in full, we see that the left hand side is equal to x multiplied by itself 6 times: x6.
    x2 × x2 × x2 = (x × x) × (x × x) × (x × x) x2 × x2 × x2 = x × x × x × x × x × x x2 × x2 × x2 = x6
Let us look at the right hand side of the equals sign (=):

x to the 2 times 3
  • x2 × 3 is a power. It consists of a base (x) raised to an exponent (2 × 3). Clearly, x2 × 3 is equal to x6. The left hand side of the equation equals the right hand side.

Lesson Slides

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See Also

What is algebra? The laws of exponents What is a power? What is a base? What is an exponent?