Negative Exponents and Negative Bases

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. \(\color{blue}{(– \frac{5x}{3yz})^{–3}} \\\ \)

Download Negative Exponents and Negative Bases Worksheet

  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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