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# Exponential Terms Raised to an Exponent

## Multiply to raise exponents to other exponents

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Practice Exponential Terms Raised to an Exponent
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Recognize and Apply the Power of a Power Property

Credit: Dan Bach
Source: https://www.flickr.com/photos/dansmath/4464539205/in/photolist-6p54Nt-6rTkFm-6p4Wpz-qfqjhD-drBzAy-6u36aG-7NzU2o-4dszrP-dZLggR-dZRY6d-dZLg4g-dZLfRV-dZLfC6-dZLfuZ-dZRX73-dZLeMe-dXZw4A-dXTQZT-dXTQxR-4F2uEv-4F2uyM-4F2ut8-4F6JbA-4F6J6q-4F6HZW-a8xZHV-7GLs7J-nhcTHN-nhnHbq-nhnuaW-nh2S78-nfgfoJ-nhinxg-nhhQEK-dkvR3Q-dkvR87-dkvR5Y-7NvVXT-8KArfj-rdLfBv-bpQyc-6nF3kc-nh3rfd-nh3HFQ-nfh4Ho-nfgwDj-nh1N6L-ngTq8S-ngToqy-quESjH

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

(x2y3z3)3

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.

### Guidance

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:

(x7y9)4

First, apply the product of a quotient property.

(x7y9)4=(x7)4(y9)4

Next, expand to simplify.

(x7)4(y9)4===x7x7x7x7y9y9y9y9x7+7+7+7y9+9+9+9x28y36

The answer is x28y36\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 a\begin{align*}a\end{align*} and b\begin{align*}b \end{align*} and any integer n\begin{align*}n\end{align*}:

(am)n=am×n

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

Simplify:

(x5)3

(x5)3==x5×3x15

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

Simplify:

(x6y3)7

(x6y3)7==x6×7y3×7x42y21

### Guided Practice

Simplify:

(x2y4z3)4

First, apply the power of a power property.

(x2y4z3)4=x2×4y4×4z3×4

Next, simplify.

x2×4y4×4z3×4=x8y16z12

The answer is x8y16z12\begin{align*}x^8y^{16}z^{12}\end{align*}.

### Examples

#### Example 1

Apply the Power of a Power Property to:

(x7)3

First, apply the power of a power property.

(x7)3=x7×3

Next, simplify.

x7×3=x21

The answer is x21\begin{align*}x^{21}\end{align*}.

#### Example 2

Apply the Power of a Power Property to:

(x3y6)3

First, apply the power of a power property.

(x3y6)3=x3×3y6×3

Next, simplify.

x3×3y6×3=x9y18

The answer is x9y18\begin{align*}x^9y^{18}\end{align*}.

#### Example 3

Apply the Power of a Power Property to:

(a7)8

First, apply the power of a power property.

(a7)8=a7×8

Next, simplify.

a7×8=a56

The answer is a56\begin{align*}a^{56}\end{align*}.

Credit: Dan Bach

Remember 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:

(x2y3z3)3

First, apply the power of a power property.

(x2y3z3)3=x2×3y3×3z3×3

Next, simplify.

x2×3y3×3z3×3=x6y9z9

The answer is x6y9z9\begin{align*}x^6y^9z^9\end{align*}.

### Explore More

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

1. (x2)2\begin{align*}(x^2)^2\end{align*}
2. (y4)3\begin{align*}(y^4)^3\end{align*}
3. (x2y4)3\begin{align*}(x^2y^4)^3\end{align*}
4. (x3y3)4\begin{align*}(x^3y^3)^4\end{align*}
5. (y6z2)6\begin{align*}(y^6z^2)^6\end{align*}
6. (x3y4)5\begin{align*}(x^3y^4)^5\end{align*}
7. (a5b3)3\begin{align*}(a^5b^3)^3\end{align*}
8. (a4b4)5\begin{align*}(a^4b^4)^5\end{align*}
9. (a3b6c7)3\begin{align*}(a^3b^6c^7)^3\end{align*}
10. (x12)3\begin{align*}(x^{12})^3\end{align*}
11. (y9)6\begin{align*}(y^9)^6\end{align*}
12. (a2b8c9)4\begin{align*}(a^2b^8c^9)^4\end{align*}
13. (x4y3z3)3\begin{align*}(x^4y^3z^3)^3\end{align*}
14. (a4b3c7d8)6\begin{align*}(a^4b^3c^7d^8)^6\end{align*}
15. (a3b11)5\begin{align*}(a^3b^{11})^5\end{align*}
16. (x6y10z12)5\begin{align*}(x^6y^{10}z^{12})^5\end{align*}

### Vocabulary Language: English

Base

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

Coefficient

A coefficient is the number in front of a variable.
Expanded Form

Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Monomial

Monomial

A monomial is an expression made up of only one term.
Power of a Power Property

Power of a Power Property

The power of a power property states that $(a^m)^n = a^{mn}$.
Variable

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.