1Cademy - Simplifying $$\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}$$ and $$\frac{y^{\frac{4}{3}} \cdot y}{y^{-\frac{2}{3}}}$$ Using the Product and Quotient Properties with Rational Exponents

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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 $\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}$ and $\frac{y^{\frac{4}{3}} \cdot y}{y^{-\frac{2}{3}}}$ Using the Product and Quotient Properties with Rational Exponents

Simplify two expressions that require both the Product Property (to combine factors in the numerator) and the Quotient Property (to divide the numerator by the denominator) when the exponents are rational. The same rules $a^m \cdot a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$ apply to fractional exponents.

ⓐ $\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}} = x^2$ :

Apply the Product Property in the numerator. Both factors share the base $x$ . Add the exponents: $x^{\frac{3}{4} + (-\frac{1}{4})} = x^{\frac{2}{4}}$ . The expression becomes $\frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}$ .
Apply the Quotient Property. Subtract the denominator exponent from the numerator exponent: $x^{\frac{2}{4} - (-\frac{6}{4})} = x^{\frac{2}{4} + \frac{6}{4}} = x^{\frac{8}{4}}$ .
Simplify. $\frac{8}{4} = 2$ , so the result is $x^2$ .

ⓑ $\frac{y^{\frac{4}{3}} \cdot y}{y^{-\frac{2}{3}}} = y^3$ :

Apply the Product Property in the numerator. The factor $y$ has an implicit exponent of $1 = \frac{3}{3}$ . Add: $y^{\frac{4}{3} + \frac{3}{3}} = y^{\frac{7}{3}}$ . The expression becomes $\frac{y^{\frac{7}{3}}}{y^{-\frac{2}{3}}}$ .
Apply the Quotient Property. Subtract: $y^{\frac{7}{3} - (-\frac{2}{3})} = y^{\frac{7}{3} + \frac{2}{3}} = y^{\frac{9}{3}}$ .
Simplify. $\frac{9}{3} = 3$ , so the result is $y^3$ .

Both examples demonstrate a three-step approach: first use the Product Property to consolidate the numerator into a single power, then use the Quotient Property to divide by the denominator, and finally simplify the resulting fractional exponent. When the denominator carries a negative exponent, the subtraction step becomes addition (subtracting a negative), which increases the overall exponent.

Updated 2026-04-21

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OpenStax Elementary Algebra

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