1Cademy - Simplifying $$x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}$$ and $$x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}$$ Using the Product Property

Learn Before

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

Example

Simplifying $x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}$ and $x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}$ Using the Product Property

Apply the Product Property for Exponents, $a^m \cdot a^n = a^{m+n}$ , to simplify expressions where the variable bases are multiplied and their rational exponents have different denominators. In these cases, find a common denominator before adding the fractions.

ⓐ $x^{\frac{1}{2}} \cdot x^{\frac{5}{6}} = x^{\frac{4}{3}}$ : Both factors share the base $x$ , so add the exponents: $x^{\frac{1}{2} + \frac{5}{6}}$ . To add the fractions, find a common denominator of $6$ . Rewrite $\frac{1}{2}$ as $\frac{3}{6}$ . The sum is $\frac{3}{6} + \frac{5}{6} = \frac{8}{6}$ . Simplify the resulting fraction by dividing the numerator and denominator by $2$ to get $\frac{4}{3}$ . The final expression is $x^{\frac{4}{3}}$ .

ⓑ $x^{\frac{1}{6}} \cdot x^{\frac{4}{3}} = x^{\frac{3}{2}}$ : Add the exponents on the common base $x$ : $x^{\frac{1}{6} + \frac{4}{3}}$ . The common denominator is $6$ . Rewrite $\frac{4}{3}$ as $\frac{8}{6}$ . The sum is $\frac{1}{6} + \frac{8}{6} = \frac{9}{6}$ . Simplify the fraction to $\frac{3}{2}$ . The final expression is $x^{\frac{3}{2}}$ .

Unlike expressions where the fractional exponents already share a denominator, these problems require fraction addition techniques before the exponent can be simplified.

Updated 2026-05-01

Contributors are:

Who are from:

References

OpenStax Intermediate Algebra

Learn Before

Related