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. 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.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.
Answer:
We have found the power of the power.Understanding Powers of a Power
Let us look at the rule for finding a power of a power, using the example above: Firstly, let us look at what is to the left of the equals sign (=):-
Inside the brackets is a power, 22. It consists of a base (2) raised to an exponent (2).
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.
(22)3 means 22 is multiplied by itself 3 times:
(22)3 = 22 × 22 × 22By 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
- 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.