Power of a Power in Math: Definition & Rule

Power of a Power in Math: Definition & Rule
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  • 0:00 What Is an Exponent?
  • 1:27 The Power of a Power Rule
  • 2:00 More Examples
  • 3:06 Lesson Summary
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Lesson Transcript
Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.

Learn ways to understand how exponent rules work when a number, variable, or algebraic expression that already has a power is raised to another power. Then test yourself with a quiz.

What Is an Exponent?

What does an exponent (otherwise known as a power) mean? For example, what does it mean when someone says 'two raised to the power of three?' It means two multiplied by itself three times, which looks like this:

2^3=2*2*2

So what's a power of a power? A power of a power means you are taking an expression that is already raised to an exponent and raising it to yet another exponent! For example, let's take 2^3 and then raise it to the fourth power, like this:

(2^3)^4

Now, the big question is. . . is there a way of combining these two exponents to make things look simpler? Simplifying gives you a much easier and neater way to state the answer to a problem. So, when it comes to a power of a power, we simplify to find an easier way to express something like (2^3)^4. To begin, let's figure out what (2^3)^4 means:

(2^3)^4=(2^3)*(2^3)*(2^3)*(2^3)

So. . . how many twos do we have? Well, remember from above what 2^3 meant? It meant 2 times itself 3 times, or 2 * 2 * 2. Let's replace that in each of our parenthesis:

(2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)

So, I ask again. . . how many twos do we have? Why don't you count them? Did you get 12 of them? Me too! We have 12 of the number 2. That means we have:

(2^3)^4=2^12

The Power of a Power Rule

Now this isn't so bad, writing out 12 of any number doesn't take too long. But what if the exponents get really big? We don't want to spend our entire day counting 2s, do we? So we created a mathematical rule that does it for us!

How would we get 12 from our previous powers of 3 and 4? Well, multiply them, of course!

(2^3)^4=2^(3*4)=2^12

To cut to the chase, the rule when taking a power to a power is such:

(a^m)^n=a^(m*n)

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