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

**At Grade**Created by: CK-12

**Practice**Exponential Terms Raised to an Exponent

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*}.**

### Video Review

Multiplying and Dividing Monomials

### Practice

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

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

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 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 .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.### Image Attributions

Here you'll recognize and apply the power of a power property.

## Concept Nodes:

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 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 .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.