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8.2: Exponent Properties Involving Quotients

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Use the quotient of powers property.
  • Use the power of a quotient property.
  • Simplify expressions involving quotient properties of exponents.


Terms introduced in this lesson:

quotient rule for exponents
power rule for quotients

Teaching Strategies and Tips

Allow students to infer in Example 1 that canceling like factors is equivalent to finding the difference in the exponents of the factors.

  • Therefore, the quotient rule for exponents is based on canceling like factors.

Additional Example:

Simplify the expression using the quotient rule.

\begin{align*}\frac{x^4} {x^3}.\end{align*}x4x3.


\begin{align*}\frac{x^4} {x^3} = \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x} {\cancel{x} \cdot \cancel{x} \cdot \cancel{x}} = \frac{x} {1} = x.\end{align*}x4x3=xxxxxxx=x1=x.

Notice also that:

\begin{align*}\frac{x^4} {x^3} = x^{4 - 3} = x^1 = x\end{align*}x4x3=x43=x1=x by the quotient rule.

Use Example 2 to show what happens when a larger exponent is subtracted from a smaller exponent.

  • Encourage students to do the division longhand.
  • Allow students to come to the conclusion that positive exponents result when subtracting smaller exponents from larger exponents.

Additional Example:

Simplify the expression leaving all powers positive.

\begin{align*}\frac{m^3n^2} {m^4n^3}.\end{align*}m3n2m4n3.

Solution: Subtract the smaller exponents from the larger exponents and leave the result in the denominator (where the larger exponents were).

\begin{align*}\frac{m^3n^2} {m^4n^3} = \frac{1} {m^{4 - 3} n^{3 - 2}} = \frac{1} {m^1n^1} = \frac{1} {mn}.\end{align*}m3n2m4n3=1m43n32=1m1n1=1mn.

Use Examples 1c, 3b, and 6 to show that the quotient rule is applied separately for each factor in the expression.

Additional Examples:

a. \begin{align*}\frac{2xy^2} {6x^3y}\end{align*}2xy26x3y

b. \begin{align*}\frac{72x^5 y^2 z} {144x^3 y^2 z^4}\end{align*}72x5y2z144x3y2z4

c. \begin{align*}\frac{56x^2 y^2 z^5} {18x^8 yz}\end{align*}56x2y2z518x8yz

Combine the quotient and power rules in Examples 5-7.

  • Suggest that students include each step in their solution.
  • Remind students to follow the order of operations.

Additional Examples:

a. \begin{align*}\left (\frac{24m^3} {10m^4}\right )^3 \cdot \left ( \frac{n^{12}} {n^5}\right )^2\end{align*}(24m310m4)3(n12n5)2

b. \begin{align*}\left (\frac{12x^2} {18y^3}\right )^2 \cdot \frac{x^3} {x^2}\end{align*}(12x218y3)2x3x2

c. \begin{align*}\left (\frac{65x^5 y^2} {145x^3 y^2}\right )^3\end{align*}(65x5y2145x3y2)3

Error Troubleshooting

In Review Question 7, have students apply the quotient rule first. This will save students some work in simplifying because the numbers will be smaller. See also Examples 4a and 4b; and Review Questions 11, 13, and 14.

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