which expression is equivalent to $81^{\frac{1}{3}}$?\n$3sqrt3{3}$\n$3sqrt{3^{3}}$\n$9sqrt3{3}$\n$27sqrt3{3}$

Explanation:

Step1: Rewrite 81 as a power of 3

Since $81 = 3^4$, then $81^{\frac{1}{3}}=(3^4)^{\frac{1}{3}}$.

Step2: Apply power - of - a - power rule

According to the rule $(a^m)^n=a^{mn}$, so $(3^4)^{\frac{1}{3}} = 3^{\frac{4}{3}}$.

Step3: Rewrite the exponent

$3^{\frac{4}{3}}=3^{1+\frac{1}{3}}$.

Step4: Use the rule $a^{m + n}=a^m\times a^n$

$3^{1+\frac{1}{3}}=3^1\times3^{\frac{1}{3}}=3\sqrt[3]{3}$.

Answer:

$3\sqrt[3]{3}$ (corresponding to the first option)