Exponential Properties Involving Quotients

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

Quotient of Powers Property: For all real numbers \\chi, \frac{x^n}{x^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. To simplify \(\begin{align*}\frac{x^7}{x^4}\end{align*}\), repeated multiplication can be used.

\( \dfrac{x^7}{x^4} = \dfrac{\not {x} \cdot \not {x} \cdot \not {x} \cdot \not {x} \cdot {x} \cdot {x} \cdot {x}}{\not {x}\cdot\not {x}\cdot\not {x}\cdot\not {x}} =\dfrac{{x} \cdot {x} \cdot{x}}{1}=x^3 \)

\( \dfrac{x^5y^3}{x^3y^2} = \dfrac{\not {x} \cdot \not {x} \not {x} \cdot {x} \cdot {x} \cdot {x}}{\not {x} \cdot\not {x} \cdot\not {x}} =\dfrac{\not {y} \cdot \not {y} \cdot y}{\not {y} \cdot \not {y}}=\dfrac{x \cdot x}{1}\cdot \dfrac{y}{1}=x^2y \)

 \( \text{OR} \ \frac{x^5y^3}{x^3y^2}=x^{5-3} \cdot y^{3-2}=x^2y \)

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

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

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

Solution:

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

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

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.

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

Example 2: Simplify the following expression.

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

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

Source: cK-12, https://www.ck12.org/book/basic-algebra/section/8.2/
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