Question

simplify. assume all variables are positive.
$(xy)^{\frac{1}{3}}$
write your answer in the form $a$ or $\frac{a}{b}$, where $a$ and $b$ are constants or variable expressions that have no variables in common. all exponents in your answer should be positive.

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For \((xy)^{\frac{1}{3}}\), we apply this rule:
\((xy)^{\frac{1}{3}} = x^{\frac{1}{3}} y^{\frac{1}{3}}\)

Step2: Rewrite with radicals (optional, but matches the form)

Recall that \(a^{\frac{1}{n}}=\sqrt[n]{a}\), so \(x^{\frac{1}{3}}=\sqrt[3]{x}\) and \(y^{\frac{1}{3}}=\sqrt[3]{y}\). Multiplying these together gives \(\sqrt[3]{x}\cdot\sqrt[3]{y}=\sqrt[3]{xy}\), but in exponential form, \(x^{\frac{1}{3}}y^{\frac{1}{3}}\) is also acceptable and meets the form requirements (since \(x^{\frac{1}{3}}\) and \(y^{\frac{1}{3}}\) have no common variables).

Answer:

\(x^{\frac{1}{3}}y^{\frac{1}{3}}\) (or \(\sqrt[3]{xy}\))