# 8.2: Exponential Properties Involving Quotients

Difficulty Level: Basic Created by: CK-12
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Practice Exponential Properties Involving Quotients

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Suppose you wanted to know the volume of a cube and the area of one of its bases. If the length of one of its edges was s\begin{align*}s\end{align*}, what would be the volume of the cube? What would be its base area? Knowing that you could find the height of the cube by dividing its volume by its base area, what expression could you write to represent this quotient? In this Concept, you'll learn about exponential properties involving quotients so that you can perform division problems such as this one.

### Guidance

In this Concept, you will learn how to simplify quotients of numbers and variables.

Quotient of Powers Property: For all real numbers χ,χnχm=χnm\begin{align*}\chi, \frac{\chi^n}{\chi^m} =\chi^{n-m}\end{align*}.

When dividing expressions with the same base, keep the base and subtract the exponent in the denominator (bottom) from the exponent in the numerator (top). When we have problems with different bases, we apply the rule separately for each base.

#### Example A

Simplify x7x4\begin{align*}\frac{x^7}{x^4}\end{align*}.

Solution: To simplify x7x4\begin{align*}\frac{x^7}{x^4}\end{align*}, repeated multiplication can be used.

x7x4x5y3x3y2=xxxxxxxxxxx=xxx1=x3=xxxxxxxxyyyyy=xx1y1=x2y OR x5y3x3y2=x53y32=x2y\begin{align*}\frac{x^7}{x^4} &= \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x}}=\frac{x \cdot x \cdot x}{1}=x^3\\ \frac{x^5y^3}{x^3y^2} &= \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x}} \cdot \frac{\cancel{y} \cdot \cancel{y} \cdot y}{\cancel{y} \cdot \cancel{y}}=\frac{x \cdot x}{1} \cdot \frac{y}{1}=x^2 y \ \text{OR} \ \frac{x^5y^3}{x^3y^2}=x^{5-3} \cdot y^{3-2}=x^2y\end{align*}

#### Example B

Simplify each of the following expressions using the quotient rule.

(a) x10x5\begin{align*}\frac{x^{10}}{x^5}\end{align*}

(b) x5γ4x3γ2\begin{align*}\frac{x^5 \gamma^4}{x^3 \gamma^2}\end{align*}

Solution:

(a) x10x5=χ105=χ5\begin{align*}\frac{x^{10}}{x^5}=\chi^{10-5}=\chi^5\end{align*}

(b) x5γ4x3γ2=χ53γ42=χ2γ2\begin{align*}\frac{x^5 \gamma^4}{x^3 \gamma^2}=\chi^{5-3} \cdot \gamma^{4-2}=\chi^2 \gamma^2\end{align*}

Power of a Quotient Property: \begin{align*}\left(\frac{\chi^n}{\gamma^m}\right)^p = \frac{\chi^{n \cdot p}}{\gamma^{m \cdot p}}\end{align*}

The power inside the parenthesis for the numerator and the denominator multiplies with the power outside the parenthesis. The situation below shows why this property is true.

#### Example C

Simplify \begin{align*}\left(\frac{x^3}{y^2}\right)^4\end{align*}.

\begin{align*}\left(\frac{x^3}{y^2}\right)^4=\left( \frac{x^3}{y^2} \right) \cdot \left( \frac{x^3}{y^2}\right) \cdot \left( \frac{x^3}{y^2} \right) \cdot \left( \frac{x^3}{y^2} \right)=\frac{(x \cdot x \cdot x) \cdot (x \cdot x \cdot x) \cdot (x \cdot x \cdot x) \cdot (x \cdot x \cdot x)}{(y \cdot y) \cdot (y \cdot y) \cdot (y \cdot y) \cdot (y \cdot y)}=\frac{x^{12}}{y^8}\end{align*}

### Vocabulary

Quotient of Powers Property: For all real numbers \begin{align*}\chi\end{align*},

\begin{align*}\frac{\chi^n}{\chi^m} =\chi^{n-m}\end{align*}.

Power of a Quotient Property:

\begin{align*}\left(\frac{\chi^n}{\gamma^m}\right)^p = \frac{\chi^{n \cdot p}}{\gamma^{m \cdot p}}\end{align*}

### Guided Practice

Simplify the following expression.

\begin{align*}\left( \frac{x^{10}}{\gamma^5} \right)^3\end{align*}

Solution:

\begin{align*}\left(\frac{x^{10}}{\gamma^5}\right)^3 = \frac{\chi^{10 \cdot 3}}{\gamma^{5 \cdot 3}} = \frac{\chi^{30}}{\gamma^{15}}\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Exponent Properties Involving Quotients (9:22)

Evaluate the following expressions.

1. \begin{align*}\frac{5^6}{5^2}\end{align*}
2. \begin{align*}\frac{6^7}{6^3}\end{align*}
3. \begin{align*}\frac{3^{10}}{3^4}\end{align*}
4. \begin{align*}\left(\frac{2^2}{3^3}\right)^3\end{align*}

Simplify the following expressions.

1. \begin{align*}\frac{a^3}{a^2}\end{align*}
2. \begin{align*}\frac{x^9}{x^5}\end{align*}
3. \begin{align*}\frac{x^{10}}{x^5}\end{align*}
4. \begin{align*}\frac{a^6}{a}\end{align*}
5. \begin{align*}\frac{a^5b^4}{a^3b^2}\end{align*}
6. \begin{align*}\frac{4^5}{4^2}\end{align*}
7. \begin{align*}\frac{5^3}{5^7}\end{align*}
8. \begin{align*}\left( \frac{3^4}{5^2} \right)^2\end{align*}
9. \begin{align*}\left( \frac{a^3b^4}{a^2b} \right)^3\end{align*}
10. \begin{align*}\frac{x^6y^5}{x^2y^3}\end{align*}
11. \begin{align*}\frac{6x^2y^3}{2xy^2}\end{align*}
12. \begin{align*}\left( \frac{2a^3b^3}{8a^7b} \right)^2\end{align*}
13. \begin{align*}(x^2)^2 \cdot \frac{x^6}{x^4}\end{align*}
14. \begin{align*}\left( \frac{16 a^2}{4b^5} \right)^3 \cdot \frac{b^2}{a^{16}}\end{align*}
15. \begin{align*}\frac{6a^3}{2a^2}\end{align*}
16. \begin{align*}\frac{15x^5}{5x}\end{align*}
17. \begin{align*}\left( \frac{18 a^{10}}{15 a^4} \right)^4\end{align*}
18. \begin{align*}\frac{25yx^6}{20 y^5 x^2}\end{align*}
19. \begin{align*}\left( \frac{x^6 y^2}{x^4y^4} \right)^3\end{align*}
20. \begin{align*}\left( \frac{6a^2}{4b^4} \right)^2 \cdot \frac{5b}{3a}\end{align*}
21. \begin{align*}\frac{(3ab)^2(4a^3b^4)^3}{(6a^2b)^4}\end{align*}
22. \begin{align*}\frac{(2a^2bc^2)(6abc^3)}{4ab^2c}\end{align*}

Mixed Review

1. Evaluate \begin{align*}x|z|-|z|\end{align*} when \begin{align*}x=8\end{align*} and \begin{align*}z=-4\end{align*}.
2. Graph the solution set to the system \begin{align*}\begin{cases} y<-x-2 \\ y \ge -6x+3 \end{cases}\end{align*}.
3. Evaluate \begin{align*}\binom{8}{4}\end{align*}.
4. Make up a situation that can be solved by 4!.
5. Write the following as an algebraic sentence: A number cubed is 8.

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### Vocabulary Language: English Spanish

TermDefinition
Power of a Quotient Property $\left(\frac{\chi^n}{\gamma^m}\right)^p = \frac{\chi^{n \cdot p}}{\gamma^{m \cdot p}}$
Quotient of Powers Property For all real numbers $\chi$, $\frac{\chi^n}{\chi^m} =\chi^{n-m}$.
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.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Power The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".

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