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# Exponential Terms Raised to an Exponent

## Multiply to raise exponents to other exponents

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Recognize and Apply the Power of a Power Property

Laura had to solve the following problem and then explain it to the class.

\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? Can you help Laura so she knows she has the right answer before explaining it to the class?

In this concept, you will learn to recognize and apply the power of a power property.

### Power of a Power Property

Exponents are a useful tool in simplifying expressions. When the rules of exponents are followed, the patterns become clear.

Let’s look at an example where you have to solve powers raised to a power.

Simplify:

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

First, apply the product of a quotient property.

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

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{(x^7)^4}{(y^9)^4} &=& \frac{x^7 \cdot x^7 \cdot x^7 \cdot x^7}{y^9 \cdot y^9 \cdot y^9 \cdot y^9} \\ &=& \frac{x^{7+7+7+7}}{y^{9+9+9+9}} \\ &=& \frac{x^{28}}{y^{36}} \end{array}\end{align*}

The answer is \begin{align*}\frac{x^{28}}{y^{36}}\end{align*}.

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 \times n}\end{align*}

Let’s look at an example of the power of a power property.

Simplify:

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

\begin{align*} \begin{array}{rcl} (x^5)^3 &=& x^{5 \times 3} \\ &=& x^{15} \end{array}\end{align*}

Let’s look at another example of the power of a power property.

Simplify:

\begin{align*}(x^6y^3)^7\end{align*}

\begin{align*}\begin{array}{rcl} (x^6y^3)^7 &=& x^{6 \times 7}y^{3 \times 7} \\ &=& x^{42}y^{21} \end{array}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Laura and her problem. Laura has to solve the following problem and then explain it to the class. She isn’t quite sure how to solve the problem. Can you help her out?

She has to apply the power of a power property.

Simplify:

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

First, apply the power of a power property.

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

Next, simplify.

\begin{align*}x^{2 \times 3}y^{3 \times 3}z^{3 \times 3}=x^6y^9z^9\end{align*}

The answer is \begin{align*}x^6y^9z^9\end{align*}.

#### Example 2

Simplify:

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

First, apply the power of a power property.

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

Next, simplify.

\begin{align*}x^{2 \times 4}y^{4 \times 4}z^{3 \times 4} = x^8y^{16}z^{12}\end{align*}

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

#### Example 3

Apply the Power of a Power Property to:

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

First, apply the power of a power property.

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

Next, simplify.

\begin{align*}x^{7 \times 3}=x^{21}\end{align*}

The answer is \begin{align*}x^{21}\end{align*}.

#### Example 4

Apply the Power of a Power Property to:

\begin{align*}(x^3y^6)^3\end{align*}

First, apply the power of a power property.

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

Next, simplify.

\begin{align*}x^{3 \times 3}y^{6 \times 3}=x^9y^{18}\end{align*}

The answer is \begin{align*}x^9y^{18}\end{align*}.

#### Example 5

Apply the Power of a Power Property to:

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

First, apply the power of a power property.

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

Next, simplify.

\begin{align*}a^{7 \times 8}=a^{56}\end{align*}

The answer is \begin{align*}a^{56}\end{align*}.

### Review

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^4y^3z^3)^3\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.