1Cademy - Simplifying $$2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}$$, $$x^{\frac{2}{3}} \cdot x^{\frac{4}{3}}$$, and $$z^{\frac{3}{4}} \cdot z^{\frac{5}{4}}$$ Using the Product Property with Rational Exponents

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Product Property for Exponents

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

Apply the Product Property for Exponents to three expressions whose exponents are fractions (rational numbers). When the bases match, add the fractional exponents just as with integer exponents — the same rule $a^m \cdot a^n = a^{m+n}$ applies when $m$ and $n$ are rational.

ⓐ $2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}} = 8$ : Both factors share the base $2$ . Add the exponents: $2^{\frac{1}{2} + \frac{5}{2}} = 2^{\frac{6}{2}}$ . Simplify the fraction: $\frac{6}{2} = 3$ , so the expression becomes $2^3 = 8$ .

ⓑ $x^{\frac{2}{3}} \cdot x^{\frac{4}{3}} = x^2$ : Both factors share the base $x$ . Add the exponents: $x^{\frac{2}{3} + \frac{4}{3}} = x^{\frac{6}{3}}$ . Simplify: $\frac{6}{3} = 2$ , giving $x^2$ .

ⓒ $z^{\frac{3}{4}} \cdot z^{\frac{5}{4}} = z^2$ : Both factors share the base $z$ . Add the exponents: $z^{\frac{3}{4} + \frac{5}{4}} = z^{\frac{8}{4}}$ . Simplify: $\frac{8}{4} = 2$ , giving $z^2$ .

In each case the procedure is the same as with integer exponents: keep the base and add the exponents. With fractional exponents, the addition step involves adding fractions — here the denominators already match, so only the numerators are added. After adding, the resulting fraction may simplify to a whole number, yielding a familiar integer exponent in the final answer. Part ⓐ shows that a numerical base with rational exponents can be fully evaluated to produce a single number, while parts ⓑ and ⓒ demonstrate that variable bases remain in exponential form with the simplified exponent.

Updated 2026-05-01

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OpenStax Elementary Algebra

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