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Power To A Power: Rule & Exponents

Instructor: DaQuita Hester

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

What does it mean to raise a power to a power? Find out here and learn about the simple rule to use when solving these problems. Then, test your understanding with a quiz.

Definition

Another word used to describe an exponent is power. So, when you hear the phrase power to a power, it just means to raise one exponent to another.

Keep in mind that exponents can come in various forms. They can be real numbers or even monomials and polynomials. No matter what form the exponent comes in, the same rule applies when calculating a power to a power. The rule is to multiply the exponents together.

Let's take a look at why this rule works. Think about what it means for something to be squared. If we square a number, that means we are multiplying that number by itself. For example, 7^2 = 7*7. The same applies if we decide to square a term like x^3. We will see that (x^3)^2 = (x^3)*( x^3).

Keeping the rules in mind for multiplying monomials, we have to add the exponents in order to simplify this problem. This helps us see that (x^3)^2 = (x^3)*( x^3) = x^6 because 3 + 3 = 6. Therefore, we can conclude that (x^3)^2 = x^6 and we notice 6 is the number we would get if we had simply multiplied the two exponents together.

Examples

Let's do a little bit of practice. To begin, let's look at (x^5)^2. To simplify this term, all we have to do is multiply the two exponents together. 5*2 = 10, so (x^5)^2 = x ^ 10.

Now, let's raise (x^3i) to the third power. This means that we want to simplify (x^3i)^3. We must still multiply the exponents together. Since 3i*3 = 9i, raising (x^3i) to the third power gives us a final answer of (x^9i).

For our next example, let's take a look at a term that has a variable as one of the exponents. Look at (x^2)^3x. We are going to follow the same process and multiply the exponents together. Since 2*3x = 6x, our term simplifies to x^6x.

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