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# Exponential Properties Involving Quotients

## Subtract exponents to divide exponents by other exponents

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Recognize and Apply the Power of a Quotient Property

Professor Smith works in a laboratory and is training a new intern, Rakesh, in a specific task. Professor Smith tells Rakesh that when she herself did the task, she started with a very small sample of cobalt. She had 10 grams of it and took one-third of one-third of one-third of one-third of it. How can Rakesh figure out how many grams of the sample Professor Smith actually ended up using?

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

### Power of a Quotient Property

Exponents can be applied to both fractions and quotients. For example, \begin{align*}\left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^4 \end{align*}. To evaluate this multiplication, find the product of the numerators and the product of the denominators.

\begin{align*}\begin{array}{rcl} \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} &=& \left( \frac{1}{2} \right)^4 \\ &=& \frac{1^4}{2^4} \\ &=& \frac{1}{16} \end{array}\end{align*}

The Power of a Quotient Property says that 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*}\left( \frac{a}{b} \right)^n =\frac{a^n}{b^n}\end{align*}

Let’s look at an example.

Simplify:\begin{align*}\left( \frac{5}{3} \right)^4\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{5}{3} \right)^4 = \frac{5^4}{3^4}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{5^4}{3^4} &=& \frac{5 \cdot 5 \cdot 5 \cdot 5}{3 \cdot 3 \cdot 3 \cdot 3} \\ &=& \frac{625}{81} \end{array}\end{align*}

The answer is \begin{align*}\frac{625}{81}\end{align*}.

Let’s look at another example.

Simplify:\begin{align*}\left( \frac{3k}{2j} \right)^4\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{3k}{2j} \right)^4 = \frac{(3k)^4}{(2j)^4}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{(3k)^4}{(2j)^4} &=& \frac{3^4 k^4}{2^2 j^4} \\ &=& \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot k \cdot k \cdot k \cdot k}{2 \cdot 2 \cdot 2 \cdot 2 \cdot j \cdot j \cdot j \cdot j} \\ &=& \frac{81k^4}{16j^4} \end{array}\end{align*}

The answer is \begin{align*}\frac{81k^4}{16j^4}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Professor Smith and the original 10 grams of cobalt.

Rakesh needs to figure out how much of the cobalt Professor Smith actually ended up using after she took one-third of one-third of one-third of one-third of 10 grams.

To figure out the number of grams in the sample, he must use monomials and powers.

First, set up the problem to model the information given in the story.

\begin{align*}10 \times \left( \frac{1}{3} \right)^4\end{align*}

Next, apply the product of a quotient property.

\begin{align*}10 \times \left( \frac{1}{3} \right)^4 = 10 \times \frac{1^4}{3^4}\end{align*}

Then, expand to simplify.

\begin{align*}\begin{array}{rcl} 10 \times \frac{1^4}{3^4} &=& 10 \times \frac{1 \cdot 1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3 \cdot 3} \\ &=& 10 \times \frac{1}{81} \\ &=& \frac{10}{81} \end{array}\end{align*}

The answer is \begin{align*} \frac{10}{81}\end{align*}.

Professor Smith’s sample size was \begin{align*}\frac{10}{81}\end{align*} grams.

#### Example 2

Simplify the following quotient.

\begin{align*}\left( \frac{-4x}{3y} \right)^3\end{align*}

First, apply the product of a quotient property.

\begin{align*}\left( \frac{-4x}{3y} \right)^3 = \frac{(-4x)^3}{(3y)^3}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{(-4x)^3}{(3y)^3} &=& \frac{-4 \cdot -4 \cdot -4 \cdot x \cdot x \cdot x}{3 \cdot 3 \cdot 3 \cdot y \cdot y \cdot y} \\ &=& \frac{-64x^3}{27y^3} \end{array}\end{align*}

The answer is \begin{align*} \frac{-64x^3}{27y^3}\end{align*}.

#### Example 3

Simplify the quotient:\begin{align*}\left( \frac{4}{5} \right)^3\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{4^3}{5^3} &=& \frac{4 \cdot 4 \cdot 4}{5 \cdot 5 \cdot 5} \\ &=& \frac{64}{125} \end{array} \end{align*}

The answer is \begin{align*}\frac{64}{125}\end{align*}.

#### Example 4

Simplify the quotient:\begin{align*}\left( \frac{2a}{3b} \right)^2\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{2a}{3b} \right)^2=\frac{2^2a^2}{3^2b^2}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{2 a^2}{3^2b^2} &=& \frac{2 \cdot 2 \cdot a \cdot a}{3 \cdot 3 \cdot b \cdot b} \\ &=& \frac{4a^2}{9b^2} \end{array} \end{align*}

The answer is \begin{align*}\frac{4a^2}{9b^2}\end{align*}.

#### Example 5

Simplify the quotient:\begin{align*}\left( \frac{a}{5b} \right)^3\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{a}{5b} \right)^3=\frac{a^3}{5^3b^3}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{a^3}{5^3b^3} &=& \frac{a \cdot a \cdot a}{5 \cdot 5 \cdot 5 \cdot b \cdot b \cdot b} \\ &=& \frac{a^3}{125b^3} \end{array}\end{align*}

The answer is \begin{align*}\frac{a^3}{125b^3}\end{align*}.

### Review

Simplify.

1. \begin{align*}\left( \frac{2}{3} \right)^4\end{align*}
2. \begin{align*}\left( \frac{1}{3} \right)^3\end{align*}
3. \begin{align*}\left( \frac{7}{8} \right)^2\end{align*}
4. \begin{align*}\left( \frac{2}{5} \right)^4\end{align*}
5. \begin{align*}\left( \frac{7k}{-2m} \right)^3 \end{align*}
6. \begin{align*}\left( \frac{3x}{-2y} \right)^3\end{align*}
7. \begin{align*}\left( \frac{4x}{-3y} \right)^4\end{align*}
8. \begin{align*}\left( \frac{5y}{-2z} \right)^5\end{align*}
9. \begin{align*}\left( \frac{-2y}{4z} \right)^4 \end{align*}
10. \begin{align*}\left( \frac{4xy}{-2z^5} \right)^5\end{align*}
11. \begin{align*}\left( \frac{12x^2y^4}{-6z^3} \right)^2\end{align*}
12. \begin{align*}\left( \frac{7x^2y}{-2z^3} \right)^3\end{align*}
13. \begin{align*}\left( \frac{2x^3y^2}{-2z^3} \right)^3 \end{align*}
14. \begin{align*}\left( \frac{x^{11}}{y^9} \right)^5\end{align*}
15. \begin{align*}\left( \frac{-5x^3}{3h^2j^8} \right)^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 The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
Power of a Product Property The power of a product property states that $(ab)^m = a^m b^m$.
Power of a Quotient Property The power of a quotient property states that $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$.
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.