# Zero and Negative Exponents Learn how to solve mathematical problems containing Zero and Negative Exponents using exponents formula.

## Step by step guide to solve zero and negative exponents problems

• A negative exponent simply means that the base is on the wrong side of the fraction line,so you need to flip the base to the other side. For instance, “$$x^{ \ -2}$$” (pronounced as “ecks to the minus two”) just means “$$x^2$$” but underneath, as in $$\frac{1}{x^2}$$.
• Any number (except zero) to the power of zero is 1. $$\color{blue}{x^{0}= {1} }$$

### Example 1:

Evaluate. $$(\frac{2x}{3}) ^{ \ 0} =$$

Solution:

Use Exponent’s rules: $$\color{blue}{x^{0}= {1} }$$

Then: $$(\frac{2x}{3}) ^{ \ 0} = { \ 1}$$

### Example 2:

Evaluate. $$\color{blue}{3^{–2}}$$

Solution:

Use Exponent’s rules: $$\color{blue}{x^{-b}= \frac{1}{x^b} } → 3^{-2}= \frac{1}{3^2}= \frac{1}{9}$$

### Example 3:

Simplify. $$(\frac{3}{2})^{-2}=$$

Solution:

Use Exponent’s rules: $$\color{blue}{(\frac{x^a}{x^b})^{-n} = (\frac{x^b}{x^a})^{n}} → {(\frac{3}{2})^{ \ -2} = (\frac{2}{3})^{2}= \frac{(2)^2}{(3)^2} = \frac{4}{9} }$$

### Example 4:

Evaluate. $$(\frac{5}{6})^{-3}=$$

Solution:

Use Exponent’s rules: $$\color{blue}{(\frac{x^a}{x^b})^{-n} = (\frac{x^b}{x^a})^{n}} → {(\frac{5}{6})^{ \ -3} = (\frac{6}{5})^{3}= \frac{(6)^3}{(5)^3} = \frac{216}{125} }$$

## Exercises

### Evaluate the following expressions.

1. $$\color{blue}{8^{–1}}$$
2. $$\color{blue}{7^{–3}}$$
3. $$\color{blue}{6^{–2}}$$
4. $$\color{blue}{(\frac{2}{3})^{–2}} \\\$$
5. $$\color{blue}{(\frac{1}{5})^{– 3}} \\\$$
6. $$\color{blue}{(\frac{1}{2})^{–8}}$$

1. $$\color{blue}{\frac{1}{8}} \\\$$
2. $$\color{blue}{ \frac{1}{343} } \\\$$
3. $$\color{blue}{ \frac{1}{36} } \\\$$
4. $$\color{blue}{ \frac{9}{4} } \\\$$
5. $$\color{blue}{125}$$
6. $$\color{blue}{256}$$ 