# 12.10: Recognize and Apply the Power of a Power Property

Difficulty Level: At Grade Created by: CK-12
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Practice Exponential Terms Raised to an Exponent

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Have you ever tried to multiply a power by a power when there is a monomial? Take a look at this dilemma.

\begin{align*}(x^2y^3z^3)^3\end{align*}

This is a monomial expression that is being raised to the third power. Do you know how to simplify this expression?

Pay attention and you will know how to complete this dilemma by the end of the Concept.

### Guidance

We have raised monomials to a power, products to a power, and quotients to a power. You can see that exponents are a useful tool in simplifying expressions. If you follow the rules of exponents, the patterns become clear. We have already seen powers taken to a power. For example, look at the quotient:

\begin{align*}\left(\frac{x^7}{y^9}\right)^4=\frac{(x^7)^4}{(y^9)^4}=\frac{(x^7)(x^7)(x^7)(x^7)}{(y^9)(y^9)(y^9)(y^9)}=\frac{x^{7+7+7+7}}{y^{9+9+9+9}}=\frac{x^{28}}{y^{36}}\end{align*}

If you focus on just the numerator, you can see that \begin{align*}(x^7)^4=x^{28}\end{align*}. You can get the exponent 28 by multiplying 7 and 4. This is an example of the Power of a Power Property which says for any nonzero numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and any integer \begin{align*}n\end{align*}:

\begin{align*}(a^m)^n=a^{m \cdot n}\end{align*}

Here is one.

\begin{align*}(x^5)^3=x^{5.3}=x^{15}\end{align*}

Take a look at this one.

\begin{align*}(x^6 y^3)^7=x^{6 \cdot 7} y^{3 \cdot 7}=x^{42} y^{21}\end{align*}

Apply the Power of a Power Property to each example.

#### Example A

\begin{align*}(x^7)^3\end{align*}

Solution: \begin{align*}x^{21}\end{align*}

#### Example B

\begin{align*}(x^3y^4)^3\end{align*}

Solution: \begin{align*}x^9y^{12}\end{align*}

#### Example C

\begin{align*}(a^7)^8\end{align*}

Solution: \begin{align*}a^{56}\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

\begin{align*}(x^2y^3z^3)^3\end{align*}

Next, we have to take each part of the monomial and raise it to the third power.

\begin{align*}(x^2)^3 = x(2 \times 3) = x^6\end{align*}

\begin{align*}(y^3)^3 = y(3 \times 3) = y^9\end{align*}

\begin{align*}(z^3)^3 = z(3 \times 3) = z^9\end{align*}

Now we can put it altogether.

\begin{align*}x^6y^9z^9\end{align*}

This is our solution.

### Vocabulary

Monomial
a single term of variables, coefficients and powers.
Coefficient
the number part of a monomial or term.
Variable
the letter part of a term
Exponent
the little number, the power, that tells you how many times to multiply the base by itself.
Base
the number that is impacted by the exponent.
Expanded Form
write out all of the multiplication without an exponent.
Power of a Power Property
the exponent is applied to all the terms inside the parentheses by multiplying the powers together.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}(x^2y^4z^3)^4\end{align*}

Solution

First, we are going to separate each part of the monomial and raise it to the fourth power.

\begin{align*}(x^2)^4 = x^8\end{align*}

\begin{align*}(y^4)^4 = y^{16}\end{align*}

\begin{align*}(z^3)^4 = z^{12}\end{align*}

Our final answer is \begin{align*}x^8y^{16}z^{12}\end{align*}.

### Practice

Directions:Simplify each monomial expression by applying the Power of a Power Property.

1. \begin{align*}(x^2)^2\end{align*}
2. \begin{align*}(y^4)^3\end{align*}
3. \begin{align*}(x^2y^4)^3\end{align*}
4. \begin{align*}(x^3y^3)^4\end{align*}
5. \begin{align*}(y^6z^2)^6\end{align*}
6. \begin{align*}(x^3y^4)^5\end{align*}
7. \begin{align*}(a^5b^3)^3\end{align*}
8. \begin{align*}(a^4b^4)^5\end{align*}
9. \begin{align*}(a^3b^6c^7)^3\end{align*}
10. \begin{align*}(x^{12})^3\end{align*}
11. \begin{align*}(y^9)^6\end{align*}
12. \begin{align*}(a^2b^8c^9)^4\end{align*}
13. \begin{align*}(x^4b^3c^3)^5\end{align*}
14. \begin{align*}(a^4b^3c^7d^8)^6\end{align*}
15. \begin{align*}(a^3b^{11})^5\end{align*}
16. \begin{align*}(x^6y^{10}z^{12})^5\end{align*}

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### Vocabulary Language: English

TermDefinition
Base When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.
Coefficient A coefficient is the number in front of a variable.
Expanded Form Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Monomial A monomial is an expression made up of only one term.
Power of a Power Property The power of a power property states that $(a^m)^n = a^{mn}$.
Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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