1Cademy - Simplifying $$\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$$ and $$\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$$ Using the Quotient Property

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Simplifying $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ , $\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$ , and $\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$ Using the Quotient Property with Rational Exponents

Example

Simplifying $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$ and $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$ Using the Quotient Property

Apply the Quotient Property for Exponents, $\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ (when the denominator's exponent is larger), to simplify rational expressions where the numerator and denominator share a variable base with fractional exponents.

ⓐ $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}} = \frac{1}{x^{\frac{4}{3}}}$ : Both terms share the base $x$ . Since the exponent in the denominator ( $\frac{5}{3}$ ) is larger than the exponent in the numerator ( $\frac{1}{3}$ ), subtract the numerator's exponent from the denominator's exponent and leave the result in the denominator: $\frac{1}{x^{\frac{5}{3} - \frac{1}{3}}}$ . Simplify the subtraction: $\frac{5}{3} - \frac{1}{3} = \frac{4}{3}$ . The simplified expression is $\frac{1}{x^{\frac{4}{3}}}$ .

ⓑ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}} = \frac{1}{x}$ : Both terms share the base $x$ , with the larger exponent in the denominator. Subtract the numerator's exponent from the denominator's exponent: $\frac{1}{x^{\frac{5}{3} - \frac{2}{3}}}$ . Simplify the subtraction: $\frac{5}{3} - \frac{2}{3} = \frac{3}{3}$ . This reduces to the integer $1$ , yielding $\frac{1}{x^1}$ , which simplifies to $\frac{1}{x}$ .

By subtracting the smaller exponent from the larger exponent directly in the denominator, you avoid negative exponents and arrive at the simplified positive-exponent form efficiently.

Updated 2026-05-01

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OpenStax Intermediate Algebra

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