Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Lesson

Notes

Negative Exponents

\(\displaystyle a^{-b}=\frac{1}{a^b}\)

Practice Problems

Express each without any negative exponents

\(\textbf{1)}\) \(9^{-1}\)

Show Answer

The answer is \(\frac{1}{9}\)

\(\textbf{2)}\) \(\left(\frac{1}{9}\right)^{-1}\)

Show Answer

The answer is \(9\)

\(\textbf{3)}\) \(9^{\frac{1}{2}}\)

Show Answer

The answer is \(3\)

\(\textbf{4)}\) \(9^{-\frac{1}{2}}\)

Show Answer

The answer is \(\frac{1}{3}\)

\(\textbf{5)}\) \(\left(\frac{1}{9}\right)^{\frac{1}{2}}\)

Show Answer

The answer is \(\frac{1}{3}\)

\(\textbf{6)}\) \(\left(\frac{1}{9}\right)^{-\frac{1}{2}}\)

Show Answer

The answer is \(3\)

\(\textbf{7)}\) \(5^{-2}\)

Show Answer

The answer is \(\frac{1}{25}\)

\(\textbf{8)}\) \(\displaystyle \left(\frac{1}{4}\right) ^{-2} \)

Show Answer

The answer is \(16\)

\(\textbf{9)}\) \(\displaystyle \left(\frac{1}{4}\right) ^{-1/2} \)

Show Answer

The answer is \(2\)

Show Work

\(\,\,\,\,\,\,\displaystyle \left(\frac{1}{4}\right) ^{-1/2}\)

\(\,\,\,\,\,\,\displaystyle \left(\frac{4}{1}\right) ^{1/2}\)

\(\,\,\,\,\,\,\displaystyle \left(4\right) ^{1/2}\)

\(\,\,\,\,\,\,\displaystyle \sqrt{4} \)

\(\,\,\,\,\,\,\displaystyle 2 \)

\(\textbf{10)}\) \(4^{-1/2}\)

Show Answer

The answer is \(\frac{1}{2}\)

\(\textbf{11)}\) \( (2z^3 w^6 )^{-4} \)

Show Answer

The answer is \( \displaystyle\frac{1}{16z^{12} w^{24}}\)

\(\textbf{12)}\) \( \displaystyle \left(\frac{4}{9}\right) ^{-1/2} \)

Show Answer

The answer is \( \displaystyle \frac{3}{2} \)

\(\textbf{13)}\) \( \displaystyle\frac{4x^4 y^{-2} z}{6x^{-1} yz} \)

Show Answer

The answer is \( \displaystyle\frac{2x^5}{3y^3} \)

\(\textbf{14)}\) \( \displaystyle \left(\frac{5x^5 y^{-3} z^2}{6x^{-2} yz} \right)^{-2} \)

Show Answer

The answer is \( \displaystyle\frac{36y^8}{25x^{14} z^2} \)

\(\textbf{15)}\) \( 3^{-4} \div 3^{-8} \)

Show Answer

The answer is \( 3^4=81 \)

Show Work

\(\)\,\,\,\,\,3^{-4} \div 3^{-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\(\)

See Related Pages\(\)

\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Zero Exponents}\)
\(\,\,\,\,\,\,\,\,3^{0}=1…\)

\(\bullet\text{ Rational Exponents}\)
\(\,\,\,\,\,\,\,\,\)

In Summary

A negative exponent indicates that the number or variable at the base is a reciprocal. For example, \(a^{-2}=\frac{1}{a^2}\).

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!