Negative exponents represent reciprocals: moving a factor across a fraction bar changes the sign of its exponent. For example, \(a^{-3}=\frac{1}{a^3}\) and \(\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n\). These problems include numerical expressions, fractional exponents, variables, products, quotients, and combinations of exponent rules.
Lesson
Notes
Negative Exponents
\(\displaystyle a^{-b}=\frac{1}{a^b}\)
Practice Problems
Express each without any negative exponents
\(\textbf{1)}\) \(9^{-1}\) The answer is \(\frac{1}{9}\)
\(\,\,\,\,\,\,9^{-1}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9^1}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9}\)
The answer is \(\frac{1}{9}\)
\(\,\,\,\,\,\,9^{-1}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9^1}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9}\)
The answer is \(\frac{1}{9}\)
\(\textbf{2)}\) \(\left(\frac{1}{9}\right)^{-1}\) The answer is \(9\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{-1}\)
\(\,\,\,\,\,\,\left(\frac{9}{1}\right)^1\)
\(\,\,\,\,\,\,9\)
The answer is \(9\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{-1}\)
\(\,\,\,\,\,\,\left(\frac{9}{1}\right)^1\)
\(\,\,\,\,\,\,9\)
The answer is \(9\)
\(\textbf{3)}\) \(9^{\frac{1}{2}}\) The answer is \(3\)
\(\,\,\,\,\,\,9^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{9}\)
\(\,\,\,\,\,\,3\)
The answer is \(3\)
\(\,\,\,\,\,\,9^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{9}\)
\(\,\,\,\,\,\,3\)
The answer is \(3\)
\(\textbf{4)}\) \(9^{-\frac{1}{2}}\) The answer is \(\frac{1}{3}\)
\(\,\,\,\,\,\,9^{-\frac{1}{2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9^{1/2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{\sqrt{9}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3}\)
The answer is \(\frac{1}{3}\)
\(\,\,\,\,\,\,9^{-\frac{1}{2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{9^{1/2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{\sqrt{9}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3}\)
The answer is \(\frac{1}{3}\)
\(\textbf{5)}\) \(\left(\frac{1}{9}\right)^{\frac{1}{2}}\) The answer is \(\frac{1}{3}\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{\frac{1}{9}}\)
\(\,\,\,\,\,\,\frac{\sqrt{1}}{\sqrt{9}}\)
\(\,\,\,\,\,\,\frac{1}{3}\)
The answer is \(\frac{1}{3}\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{\frac{1}{9}}\)
\(\,\,\,\,\,\,\frac{\sqrt{1}}{\sqrt{9}}\)
\(\,\,\,\,\,\,\frac{1}{3}\)
The answer is \(\frac{1}{3}\)
\(\textbf{6)}\) \(\left(\frac{1}{9}\right)^{-\frac{1}{2}}\) The answer is \(3\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{-\frac{1}{2}}\)
\(\,\,\,\,\,\,\left(\frac{9}{1}\right)^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{9}\)
\(\,\,\,\,\,\,3\)
The answer is \(3\)
\(\,\,\,\,\,\,\left(\frac{1}{9}\right)^{-\frac{1}{2}}\)
\(\,\,\,\,\,\,\left(\frac{9}{1}\right)^{\frac{1}{2}}\)
\(\,\,\,\,\,\,\sqrt{9}\)
\(\,\,\,\,\,\,3\)
The answer is \(3\)
\(\textbf{7)}\) \(5^{-2}\) The answer is \(\frac{1}{25}\)
\(\,\,\,\,\,\,5^{-2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{5^2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{25}\)
The answer is \(\frac{1}{25}\)
\(\,\,\,\,\,\,5^{-2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{5^2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{25}\)
The answer is \(\frac{1}{25}\)
\(\textbf{8)}\) \(\displaystyle \left(\frac{1}{4}\right)^{-2}\) The answer is \(16\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{1}{4}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{1}\right)^2\)
\(\,\,\,\,\,\,4^2\)
\(\,\,\,\,\,\,16\)
The answer is \(16\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{1}{4}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{1}\right)^2\)
\(\,\,\,\,\,\,4^2\)
\(\,\,\,\,\,\,16\)
The answer is \(16\)
\(\textbf{9)}\) \(\displaystyle \left(\frac{1}{4}\right)^{-1/2}\) The answer is \(2\)
\(\,\,\,\,\,\,\displaystyle \left(\frac{1}{4}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle \left(\frac{4}{1}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle 4^{1/2}\)
\(\,\,\,\,\,\,\displaystyle \sqrt{4}\)
\(\,\,\,\,\,\,\displaystyle 2\)
The answer is \(2\)
\(\,\,\,\,\,\,\displaystyle \left(\frac{1}{4}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle \left(\frac{4}{1}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle 4^{1/2}\)
\(\,\,\,\,\,\,\displaystyle \sqrt{4}\)
\(\,\,\,\,\,\,\displaystyle 2\)
The answer is \(2\)
\(\textbf{10)}\) \(4^{-1/2}\) The answer is \(\frac{1}{2}\)
\(\,\,\,\,\,\,4^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{4^{1/2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{\sqrt{4}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}\)
The answer is \(\frac{1}{2}\)
\(\,\,\,\,\,\,4^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{4^{1/2}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{\sqrt{4}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}\)
The answer is \(\frac{1}{2}\)
\(\textbf{11)}\) \((2z^3w^6)^{-4}\)
The answer is \(\displaystyle\frac{1}{16z^{12}w^{24}}\)
\(\,\,\,\,\,\,(2z^3w^6)^{-4}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{(2z^3w^6)^4}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2^4z^{12}w^{24}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{16z^{12}w^{24}}\)
The answer is \(\displaystyle\frac{1}{16z^{12}w^{24}}\)
![Link to Youtube Video Solving Question Number 11](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
\(\,\,\,\,\,\,(2z^3w^6)^{-4}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{(2z^3w^6)^4}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2^4z^{12}w^{24}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{16z^{12}w^{24}}\)
The answer is \(\displaystyle\frac{1}{16z^{12}w^{24}}\)
\(\textbf{12)}\) \(\displaystyle\left(\frac{4}{9}\right)^{-1/2}\) The answer is \(\displaystyle\frac{3}{2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{9}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{9}{4}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{\frac{9}{4}}\)
\(\,\,\,\,\,\,\displaystyle\frac{3}{2}\)
The answer is \(\displaystyle\frac{3}{2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{9}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{9}{4}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{\frac{9}{4}}\)
\(\,\,\,\,\,\,\displaystyle\frac{3}{2}\)
The answer is \(\displaystyle\frac{3}{2}\)
\(\textbf{13)}\) \(\displaystyle\frac{4x^4y^{-2}z}{6x^{-1}yz}\) The answer is \(\displaystyle\frac{2x^5}{3y^3}\)
\(\,\,\,\,\,\,\displaystyle\frac{4x^4y^{-2}z}{6x^{-1}yz}\)
\(\,\,\,\,\,\,\displaystyle\frac{4}{6}x^{4-(-1)}y^{-2-1}z^{1-1}\)
\(\,\,\,\,\,\,\displaystyle\frac{2}{3}x^5y^{-3}\)
\(\,\,\,\,\,\,\displaystyle\frac{2x^5}{3y^3}\)
The answer is \(\displaystyle\frac{2x^5}{3y^3}\)
\(\,\,\,\,\,\,\displaystyle\frac{4x^4y^{-2}z}{6x^{-1}yz}\)
\(\,\,\,\,\,\,\displaystyle\frac{4}{6}x^{4-(-1)}y^{-2-1}z^{1-1}\)
\(\,\,\,\,\,\,\displaystyle\frac{2}{3}x^5y^{-3}\)
\(\,\,\,\,\,\,\displaystyle\frac{2x^5}{3y^3}\)
The answer is \(\displaystyle\frac{2x^5}{3y^3}\)
\(\textbf{14)}\) \(\displaystyle\left(\frac{5x^5y^{-3}z^2}{6x^{-2}yz}\right)^{-2}\) The answer is \(\displaystyle\frac{36y^8}{25x^{14}z^2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^5y^{-3}z^2}{6x^{-2}yz}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^7y^{-4}z}{6}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^7z}{6y^4}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{6y^4}{5x^7z}\right)^2\)
\(\,\,\,\,\,\,\displaystyle\frac{36y^8}{25x^{14}z^2}\)
The answer is \(\displaystyle\frac{36y^8}{25x^{14}z^2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^5y^{-3}z^2}{6x^{-2}yz}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^7y^{-4}z}{6}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{5x^7z}{6y^4}\right)^{-2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{6y^4}{5x^7z}\right)^2\)
\(\,\,\,\,\,\,\displaystyle\frac{36y^8}{25x^{14}z^2}\)
The answer is \(\displaystyle\frac{36y^8}{25x^{14}z^2}\)
\(\textbf{15)}\) \(3^{-4}\div3^{-8}\) The answer is \(3^4=81\)
\(\,\,\,\,\,\,3^{-4}\div3^{-8}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3^4}\div\frac{1}{3^8}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3^4}\times\frac{3^8}{1}\)
\(\,\,\,\,\,\,\displaystyle\frac{3^8}{3^4}\)
\(\,\,\,\,\,\,3^{8-4}\)
\(\,\,\,\,\,\,3^4=81\)
The answer is \(81\)
\(\,\,\,\,\,\,3^{-4}\div3^{-8}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3^4}\div\frac{1}{3^8}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{3^4}\times\frac{3^8}{1}\)
\(\,\,\,\,\,\,\displaystyle\frac{3^8}{3^4}\)
\(\,\,\,\,\,\,3^{8-4}\)
\(\,\,\,\,\,\,3^4=81\)
The answer is \(81\)
\(\textbf{16)}\) \(x^{-5}\) The answer is \(\displaystyle\frac{1}{x^5}\)
\(\,\,\,\,\,\,x^{-5}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{x^5}\)
The answer is \(\displaystyle\frac{1}{x^5}\)
\(\,\,\,\,\,\,x^{-5}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{x^5}\)
The answer is \(\displaystyle\frac{1}{x^5}\)
\(\textbf{17)}\) \(\displaystyle\frac{3a^{-2}b^3}{6ab^{-1}}\) The answer is \(\displaystyle\frac{b^4}{2a^3}\)
\(\,\,\,\,\,\,\displaystyle\frac{3a^{-2}b^3}{6ab^{-1}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}a^{-2-1}b^{3-(-1)}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}a^{-3}b^4\)
\(\,\,\,\,\,\,\displaystyle\frac{b^4}{2a^3}\)
The answer is \(\displaystyle\frac{b^4}{2a^3}\)
\(\,\,\,\,\,\,\displaystyle\frac{3a^{-2}b^3}{6ab^{-1}}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}a^{-2-1}b^{3-(-1)}\)
\(\,\,\,\,\,\,\displaystyle\frac{1}{2}a^{-3}b^4\)
\(\,\,\,\,\,\,\displaystyle\frac{b^4}{2a^3}\)
The answer is \(\displaystyle\frac{b^4}{2a^3}\)
\(\textbf{18)}\) \((x^{-2}y^3)^{-2}\) The answer is \(\displaystyle\frac{x^4}{y^6}\)
\(\,\,\,\,\,\,(x^{-2}y^3)^{-2}\)
\(\,\,\,\,\,\,x^{(-2)(-2)}y^{3(-2)}\)
\(\,\,\,\,\,\,x^4y^{-6}\)
\(\,\,\,\,\,\,\displaystyle\frac{x^4}{y^6}\)
The answer is \(\displaystyle\frac{x^4}{y^6}\)
\(\,\,\,\,\,\,(x^{-2}y^3)^{-2}\)
\(\,\,\,\,\,\,x^{(-2)(-2)}y^{3(-2)}\)
\(\,\,\,\,\,\,x^4y^{-6}\)
\(\,\,\,\,\,\,\displaystyle\frac{x^4}{y^6}\)
The answer is \(\displaystyle\frac{x^4}{y^6}\)
\(\textbf{19)}\) \(\displaystyle\left(\frac{25}{4}\right)^{-1/2}\) The answer is \(\displaystyle\frac{2}{5}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{25}{4}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{25}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{\frac{4}{25}}\)
\(\,\,\,\,\,\,\displaystyle\frac{2}{5}\)
The answer is \(\displaystyle\frac{2}{5}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{25}{4}\right)^{-1/2}\)
\(\,\,\,\,\,\,\displaystyle\left(\frac{4}{25}\right)^{1/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{\frac{4}{25}}\)
\(\,\,\,\,\,\,\displaystyle\frac{2}{5}\)
The answer is \(\displaystyle\frac{2}{5}\)
\(\textbf{20)}\) \(\displaystyle\frac{8x^{-3}y^2}{2x^2y^{-4}}\) The answer is \(\displaystyle\frac{4y^6}{x^5}\)
\(\,\,\,\,\,\,\displaystyle\frac{8x^{-3}y^2}{2x^2y^{-4}}\)
\(\,\,\,\,\,\,4x^{-3-2}y^{2-(-4)}\)
\(\,\,\,\,\,\,4x^{-5}y^6\)
\(\,\,\,\,\,\,\displaystyle\frac{4y^6}{x^5}\)
The answer is \(\displaystyle\frac{4y^6}{x^5}\)
\(\,\,\,\,\,\,\displaystyle\frac{8x^{-3}y^2}{2x^2y^{-4}}\)
\(\,\,\,\,\,\,4x^{-3-2}y^{2-(-4)}\)
\(\,\,\,\,\,\,4x^{-5}y^6\)
\(\,\,\,\,\,\,\displaystyle\frac{4y^6}{x^5}\)
The answer is \(\displaystyle\frac{4y^6}{x^5}\)
See Related Pages\(\)
\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Zero Exponents}\)
\(\,\,\,\,\,\,\,\,3^{0}=1…\)
\(\bullet\text{ Rational Exponents}\)
\(\,\,\,\,\,\,\,\,\)