Power of a Power Rule: Formula and 7 Examples

Before you start taking algebra, you may have learned about the order of operations, PEMDAS (Please Excuse My Dear Aunt Sally). The “E,” the second letter in that acronym, is the exponent. An exponent describes when a base, which can be either a number or variable, is raised to a power and multiplied by itself in some form. There are several rules associated with exponents, and today, we will be focusing on the Power of a Power rule.



As seen in the image above, the Power of a Power Rule comes into play when multiple powers interact. In this case though, unlike other rules, you do not need more than one base to distribute. Typically, with a power of a power, you have a base raised to power within the parentheses. Then on the outside of the parenthesis, there is another power. What happens here is just like in the distributive property, where the outside number will multiply into whatever is inside the parenthesis. The power on the outside will multiply with the power within. Any coefficient present will be raised to that power as well.


Example 1

As seen in this problem, the power rule is applied with exponents, but they need not be actual numbers. In this case, we have C as a coefficient multiplied by our variable x. The x is being raised to the power of b. However, this entire function is being raised to the power of a. So, in order to simplify this equation, the power amultiplies by power b and our constant C is raised to the power of b.


Example 2

With this problem, I have a function enclosed and raised to the power of y. Like in the previous question, I raise the power of the constant and multiply the powers and simplify to find the correct answer.


Example 3

With this expression, I have a constant of 3, with variable c raised to the power of 5. Outside the parentheses, it is raised to the power of 2. So, I square the constant and follow through with the power of the power rule to get my answer of 9x10.


Example 4

Here we have a variable in the power being raised, but the same rules still apply. Inside, we have 5x raised to the power of y. Outside, we have this raised to the power of 3. The same rule applies to this expression, and we get the answer.


Example 5

With this question, there 7L2 raised to the power of 3. The same procedure applies where the coefficient is raised to the power of three and the powers multiply by each other to get the correct answer.


Example 6

For this one, their 10x8 is raised to the power of y. Even with having multiple variables, keep in mind we are just simplifying the expression so we will repeat the earlier steps of distributing and multiplying the power accordingly.


Example 7

For this final one, we have 6ax raised to the power of 4. Again, the coefficient is raised to that exponent, but the powers will multiply.


Here, I have only shown how the power of a power rule applies to single-term expressions. Once you encounter binomial or trinomial expressions, the power of a power rule no longer applies. You must use a separate method known as FOIL. However, as long as you have one term being raised to a power, whether you have it raised to a variable or a number, the power of a power rule will be the method you can always turn to.


Author: Maaida Kirmani



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