Learn how to solve math problems containing negative exponents and negative bases.

## Related Topics

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

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

- Make the power positive. A negative exponent is the reciprocal of that number with a positive exponent.
- The parenthesis is important! \(-5^{ \ -2}\) is not the same as \((– 5)^{ \ -2}\)

\(– 5^{ \ -2}= -\frac{1}{5^2} \) and \((–5)^{ \ -2}=+\frac{1}{5^2}\)

### Negative Exponents and Negative Bases – Example 1:

Simplify. \((\frac{5a}{6c})^{ \ -2}=\)

**Solution:**

Use Exponent’s rules: \(\color{blue}{(\frac{x^a}{x^b})^{-n} = (\frac{x^b}{x^a})^{n}} → {(\frac{5a}{6c})^{ \ -2} = (\frac{6c}{5a})^{2}= \frac{(6c)^2}{(5a)^2} = \frac{6^2 c^2}{5^2a^2}= \frac{36 c^2}{25a^2} }\)

### Negative Exponents and Negative Bases – Example 2:

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

**Solution:**

Use Exponent’s rules: \(\color{blue}{(\frac{x^a}{x^b})^{-n} = (\frac{x^b}{x^a})^{n}} → {(\frac{2x}{3yz})^{ \ -3} = (\frac{3yz}{2x})^{3}= \frac{(3yz)^3}{(2x)^3} = \frac{3^3 y^3z^3}{2^3x^3}= \frac{27 y^3z^3}{8x^3} }\)

### Negative Exponents and Negative Bases – Example 3:

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

**Solution:**

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

### Negative Exponents and Negative Bases – Example 4:

Simplify. \((-\frac{5x}{3yz})^{-3}=\)

**Solution:**

Use Exponent’s rules: \(\color{blue}{\frac{1}{x^b} =x^{-b}} →(-\frac{5x}{3yz})^{-3}= \frac{ 1}{(-\frac{5x}{3yz})^3} = \frac{ 1}{-\frac{5^3 x^3}{3^3 y^3 z^3} }\)

Now use fraction rule: \(\color{blue}{\frac{1}{(\frac{b}{c})}=\frac{c}{b }} → \frac{ 1}{\frac{ 5^3 x^3 }{ 3^3 y^3 z^3 } } = -\frac{ 3^3 y^3 z^3 }{ 5^3 x^3 }\)

Then: \( -\frac{ 27 y^3 z^3}{125x^3}\)

## Exercises for Solveing Negative Exponents and Negative Bases

### Simplify.

- \(\color{blue}{\frac{4ab^{-2}}{-3c^{-2}} } \\\ \)
- \(\color{blue}{– 12x^2y^{-3} } \\\ \)
- \(\color{blue}{(– \frac{1}{3})^{–2}} \\\ \)
- \(\color{blue}{(– \frac{3}{4})^{–2}} \\\ \)
- \(\color{blue}{(\frac{5x}{4y})^{–2}} \\\ \)
- \(\color{blue}{(– \frac{5x}{3yz})^{–3}} \\\ \)

### Download Negative Exponents and Negative Bases Worksheet

- \(\color{blue}{– \frac{4ac^2}{3b^2} } \\\ \)
- \(\color{blue}{– \frac{12x^2}{y^3 }} \\\ \)
- \(\color{blue}{9} \\\ \)
- \(\color{blue}{\frac{16}{9}} \\\ \)
- \(\color{blue}{\frac{16y^2}{25x^2 }} \\\ \)
- \(\color{blue}{– \frac{27y^3 z^3}{125x^3 }} \\\ \)