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

## Related Topics

- How to Solve Powers of Products and Quotients
- How to Multiply Exponents
- How to Solve Negative Exponents and Negative Bases
- How to Solve Zero and Negative Exponents
- How to Solve Scientific Notation

## 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:**

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: \(\color{blue}{\frac{x^a}{x^b} =x^{a–b}}\)

\(\ →\frac{x^3}{x^2 }=x^{3–2}=x^1=x\)

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

### 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.

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

### Download Division Property of Exponents Worksheet

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