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Finding a Power of a Power in Algebra

(KS3, Year 7)

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.

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). 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. (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

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