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

Finding a Power of a Power in Algebra

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.

An Example Question

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

x 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.

x to the 6

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 bracket 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.

A Real Example of How to Find a Power of a Power in Algebra

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

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