How to Divide Exponents? (+FREE Worksheet!)

Learn how to divide exponents by using the division property of exponents in a few simple steps.

How to Divide Exponents? (+FREE Worksheet!)

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Step by step guide to divide exponents

  • For division of exponents use these formulas: \(\color{blue}{\frac{x^a}{x^b} =x^{a–b} , x≠0}\)
    \(\color{blue}{\frac{x^a}{x^b} =\frac{1}{x^{ \ b-a}} , x≠0, \frac{1}{x^b} =x^{ \ -b}}\)

Division Property of Exponents – Example 1:

Divide. \(\frac{4x^{5}}{8x^{2}} =\)

Solution:

First cancel the common factor: \(4→\frac{4x^5}{8x^2}=\frac{x^5}{2x^2}\)
Use Exponent’s rules: \(\color{blue}{\frac{x^a}{x^b} =x^{a–b}}\)

\(\ →\frac{x^5}{x^2 }=x^{5–2}=x^3\)

Then: \(\frac{4x^{5}}{8x^{2}} =\frac{x^3}{2}\)

Division Property of Exponents – Example 2:

Divide. \(\frac{18x^{ \ -6}}{2x^{ \ -3 }}=\)

Solution:

First cancel the common factor: \(2 → \frac{18x^{ \ -6}}{2x^{ \ -3 }}=\frac{9x^{ \ -6}}{x^{ \ -3 }}\)

Use Exponent’s rules: \(\color{blue}{ \frac{x^a}{x^b} =\frac{1}{x^{b-a}} } \)

\(→\frac{x^{-6}}{x^{-3}} =\frac{1}{x^{-3-(-6)}} =\frac{1}{x^{-3+6}} =\frac{1}{x^3} \)
Then: \(\frac{18x^{-6}}{2x^{-3}} =\frac{9}{x^3}\)

Division Property of Exponents – Example 3:

Simplify. \(\frac{4x^3 y}{36x^2 y^ {3}}=\)

Solution:

First cancel the common factor: \(4→\frac{4x^3 y}{36x^2 y^3 }=\frac{x^3 y}{9x^2 y^3 }\)

Use Exponent’s rules: \(\color{blue}{\frac{x^a}{x^b} =x^{a–b}}\)

\(\ →\frac{x^3}{x^2 }=x^{3–2}=x^1=x\) , \(\frac{y}{y^3 }=\frac{1}{y^{3–1} }= \frac{1}{y^{2} }\)

Then: \(\frac{4x^3 y}{36x^2 y^3 }=\frac{x}{9y^2 }\)

Division Property of Exponents – Example 4:

Divide. \(\frac{2x^{-5}}{9x^{-2}} =\)

Solution:

Use Exponent’s rules: \(\color{blue}{ \frac{x^a}{x^b} =\frac{1}{x^{b-a}} } \)

\(→\frac{x^{-5}}{x^{-2} }=\frac{1}{x^{-2-(-5)}} =\frac{1}{x^{-2+5}} =\frac{1}{x^3}\)
Then: \(\frac{2x^{-5}}{9x^{-2}} =\frac{2}{9x^3 }\)

Exercises for Dividing Property of Exponents

Simplify.

  1. \(\color{blue}{\frac{5^5}{5}} \\\ \)
  2. \(\color{blue}{\frac{3}{3^5 }} \\\ \)
  3. \(\color{blue}{\frac{12x^4}{15x^7 y^9 }} \\\ \)
  4. \(\color{blue}{\frac{12yx^4}{10yx^8}} \\\ \)
  5. \(\color{blue}{\frac{16x^4 y}{9x^8 y^2}} \\\ \)
  6. \(\color{blue}{\frac{5x^8}{20x^8 }} \\\ \)

Download Division Property of Exponents Worksheet

  1. \(\color{blue}{5^4}\)
  2. \(\color{blue}{\frac{1}{3^4}} \\\ \)
  3. \(\color{blue}{\frac{4}{5x^3 y^9 }} \\\ \ \)
  4. \(\color{blue}{\frac{6}{5x^4 }} \\\ \ \)
  5. \(\color{blue}{\frac{16}{9x^4 y}} \\\ \ \)
  6. \(\color{blue}{\frac{1}{4}} \\\ \ \)

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