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Exponential Properties Involving Quotients

Subtract exponents to divide exponents by other exponents

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Exponential Properties Involving Quotients

Suppose you have a cube with the length of one of its being \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?

Exponential Properties Involving Quotients

Quotients of Powers

The Quotient of Powers Property states that for all real numbers \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.

Let's simplify the following expressions using the Quotient of Powers Property:

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

To simplify \begin{align*}\frac{x^7}{x^4}\end{align*}, 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*}

1. \begin{align*}\frac{x^{10}}{x^5}\end{align*}

\begin{align*}\frac{x^{10}}{x^5}=\chi^{10-5}=\chi^5\end{align*}

1.  \begin{align*}\frac{x^5 \gamma^4}{x^3 \gamma^2}\end{align*}

\begin{align*}\frac{x^5 \gamma^4}{x^3 \gamma^2}=\chi^{5-3} \cdot \gamma^{4-2}=\chi^2 \gamma^2\end{align*}

Powers of Quotients

The Power of a Quotient Property states that \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.

Let's 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*}

Examples

Example 1

Earlier, you were asked what the area of one base of a cube is and what the volume of a cube is if the length of one of its edges is length \begin{align*}s\end{align*}. What is the height of the cube?

Since the base of a cube is a square, the area of that base is \begin{align*}s^2\end{align*}. The volume of a cube is \begin{align*}s^3\end{align*}. To find the height, divide the volume by the area of the base and apply the Quotient of Powers Property:

\begin{align*}\frac{s^3}{s^2}=s^{3-2}=s\end{align*}

The height, as expected, is \begin{align*}s\end{align*}.

Example 2

Simplify the following expression.

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

\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*}

Review

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.

To see the Review answers, open this PDF file and look for section 8.2.

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Color Highlighted Text Notes

Vocabulary Language: English Spanish

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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