logo
How it worksCoursesResearch CommunitiesBenefitsAbout Us
Schedule Demo
Learn Before
Example

Simplifying m23⋅m−13m−53\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}} and (25m16n116m23n−16)12\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}} Using Exponent Properties

Apply the properties of exponents to simplify these two rational expressions involving fractional exponents.

ⓐ m23⋅m−13m−53=m2\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}} = m^2:

  1. Apply the Product Property in the numerator. Add the exponents: m23+(−13)=m13m^{\frac{2}{3} + \left(-\frac{1}{3}\right)} = m^{\frac{1}{3}}. The expression becomes m13m−53\frac{m^{\frac{1}{3}}}{m^{-\frac{5}{3}}}.
  2. Apply the Quotient Property. Subtract the denominator's exponent from the numerator's exponent: m13−(−53)=m13+53=m63m^{\frac{1}{3} - \left(-\frac{5}{3}\right)} = m^{\frac{1}{3} + \frac{5}{3}} = m^{\frac{6}{3}}.
  3. Simplify. 63=2\frac{6}{3} = 2, so the result is m2m^2.

ⓑ (25m16n116m23n−16)12=5nm14\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}} = \frac{5n}{m^{\frac{1}{4}}}:

  1. Apply the Quotient Property inside the parentheses. Subtract exponents for like bases:
  • For mm: 16−23=16−46=−36=−12\frac{1}{6} - \frac{2}{3} = \frac{1}{6} - \frac{4}{6} = -\frac{3}{6} = -\frac{1}{2}
  • For nn: 116−(−16)=116+16=126=2\frac{11}{6} - \left(-\frac{1}{6}\right) = \frac{11}{6} + \frac{1}{6} = \frac{12}{6} = 2 The expression becomes (25m−12n2)12(25 m^{-\frac{1}{2}} n^2)^{\frac{1}{2}}.
  1. Apply the Product to a Power Property. Distribute the outer exponent: 2512⋅(m−12)12⋅(n2)1225^{\frac{1}{2}} \cdot (m^{-\frac{1}{2}})^{\frac{1}{2}} \cdot (n^2)^{\frac{1}{2}}.
  2. Simplify each factor.
  • 2512=25=525^{\frac{1}{2}} = \sqrt{25} = 5
  • (m−12)12=m−12⋅12=m−14(m^{-\frac{1}{2}})^{\frac{1}{2}} = m^{-\frac{1}{2} \cdot \frac{1}{2}} = m^{-\frac{1}{4}}
  • (n2)12=n2⋅12=n1=n(n^2)^{\frac{1}{2}} = n^{2 \cdot \frac{1}{2}} = n^1 = n The result is 5m−14n5 m^{-\frac{1}{4}} n.
  1. Rewrite with positive exponents. Move m−14m^{-\frac{1}{4}} to the denominator to obtain 5nm14\frac{5n}{m^{\frac{1}{4}}}

0

1

Updated 2026-05-01

Contributors are:

Claude Opus
Claude Opus
🏆 2

Who are from:

Claude
Claude
🏆 2

References

Tags

OpenStax

Intermediate Algebra @ OpenStax

Ch.8 Roots and Radicals - Intermediate Algebra @ OpenStax

Algebra

Related