# Division Property of Exponents

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

## Step by step guide to divide exponents

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

### 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: $$\frac{4x^{5}}{8x^{2}} =\frac{x^3}{2}$$

### 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: $$\frac{18x^{-6}}{2x^{-3}} =\frac{9}{x^3}$$

### 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: $$y→\frac{xy}{9y^3 }=\frac{x}{9y^2 }$$

### 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

### 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 }} \\\$$

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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