Negative Exponents

High School

Definition

Another way to display reciprocals. Negative exponents allow you to write powers without having to use decimals or fractions. When you have a negative exponent, flipping it to the other side of the division line allows you to rewrite it as a number with a positive exponent.

Worked examples

\(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

A negative exponent in the numerator flips to the denominator and becomes positive.

\(\frac{1}{3^{-4}} = 3^4 = 81\)

A negative exponent in the denominator flips to the numerator and becomes positive.

\(x^{-3} \cdot x^5 = x^{-3+5} = x^2\)

Combine exponents by adding; a negative exponent follows the same exponent rules.

Common mistakes

\(4^{-2} = -16\) → \(4^{-2} = \frac{1}{16}\) The negative exponent means reciprocal, not a negative answer.
\(\frac{1}{2^{-3}} = \frac{1}{8}\) → \(\frac{1}{2^{-3}} = 2^3 = 8\) Flipping a negative exponent from denominator to numerator makes it positive.
\(x^{-1} = -x\) → \(x^{-1} = \frac{1}{x}\) Negative exponent means reciprocal, not negating the base.

Where you'll use it next

You'll use negative exponents when simplifying rational expressions, working with scientific notation for very small numbers, and solving exponential equations in algebra and beyond.

Found in 2 StudyPug lessons

Mastering Negative Exponents: Rules, Examples, and Applications

Grade 12 Math

Unlock the power of negative exponents! Learn essential rules, work through step-by-step examples, and discover real-world applications. Elevate your math skills and conquer complex problems with confidence.

12th Grade12th

Mastering the Negative Exponent Rule in Algebra

Grade 10 Math

Unlock the power of negative exponents! Learn to simplify complex expressions, solve equations with confidence, and apply this crucial skill to real-world math problems. Boost your algebra prowess today!

10th Grade10th