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8.2: Exponential Properties Involving Quotients

Difficulty Level: At Grade Created by: CK-12

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

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

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

\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*}

x7x4x5y3x3y2=xxxxxxxxxxx=xxx1=x3=xxxxxxxxyyyyy=xx1y1=x2y OR x5y3x3y2=x53y32=x2y

Example 1: Simplify each of the following expressions using the quotient rule.

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

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


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

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

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*}(χnγm)p=χnpγmp

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.

\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*}


Example 2: 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*}(x10γ5)3=χ103γ53=χ30γ15

Practice Set

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*}5652
  2. \begin{align*}\frac{6^7}{6^3}\end{align*}6763
  3. \begin{align*}\frac{3^{10}}{3^4}\end{align*}31034
  4. \begin{align*}\left(\frac{2^2}{3^3}\right)^3\end{align*}(2233)3

Simplify the following expressions.

  1. \begin{align*}\frac{a^3}{a^2}\end{align*}a3a2
  2. \begin{align*}\frac{x^9}{x^5}\end{align*}x9x5
  3. \begin{align*}\frac{x^{10}}{x^5}\end{align*}x10x5
  4. \begin{align*}\frac{a^6}{a}\end{align*}a6a
  5. \begin{align*}\frac{a^5b^4}{a^3b^2}\end{align*}a5b4a3b2
  6. \begin{align*}\frac{4^5}{4^2}\end{align*}4542
  7. \begin{align*}\frac{5^7}{5^3}\end{align*}5753
  8. \begin{align*}\left( \frac{3^4}{5^2} \right)^2\end{align*}(3452)2
  9. \begin{align*}\left( \frac{a^3b^4}{a^2b} \right)^3\end{align*}(a3b4a2b)3
  10. \begin{align*}\frac{x^6y^5}{x^2y^3}\end{align*}x6y5x2y3
  11. \begin{align*}\frac{6x^2y^3}{2xy^2}\end{align*}
  12. \begin{align*}\left( \frac{8a^7b^3}{2a^3b} \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^6}{4b^5} \right)^3 \cdot \frac{b^{16}}{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{25y^5x^6}{20 yx^2}\end{align*}
  19. \begin{align*}\left( \frac{x^6 y^4}{x^4y^2} \right)^3\end{align*}
  20. \begin{align*}\left( \frac{6a^2}{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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