Negative Exponents and Negative Bases

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

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 $$(–5)^{ \ -2}=+\frac{1}{5^2}$$

Example 1:

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

Solution:

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

Example 2:

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

Solution:

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

Example 3:

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

Solution:

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

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

Simplify.

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

1. $$\color{blue}{– \frac{4ac^2}{3b^2} } \\\$$
2. $$\color{blue}{– \frac{12x^2}{y^3 }} \\\$$
3. $$\color{blue}{9} \\\$$
4. $$\color{blue}{\frac{16}{9}} \\\$$
5. $$\color{blue}{\frac{16y^2}{25x^2 }} \\\$$
6. $$\color{blue}{– \frac{27y^3 z^3}{125x^3 }} \\\$$

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