Question

which expression is equivalent to \\(\sqrt4{3^{3}}\\)? \\(4^{3}\\) \\(3^{\frac{3}{4}}\\) \\(3^{\frac{4}{3}}\\) \\(4^{9}\\)

Explanation:

Step1: Recall the radical - exponent conversion rule

The rule for converting a radical to a rational exponent is $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$, where $n$ is the index of the radical and $m$ is the exponent of the base inside the radical.

Step2: Apply the rule to the given expression

In the given expression $\sqrt[4]{3^{3}}$, we have $n = 4$ (the index of the fourth - root) and $m = 3$ (the exponent of the base 3 inside the radical). Using the rule $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$, when $a = 3$, $m = 3$, and $n = 4$, we get $\sqrt[4]{3^{3}}=3^{\frac{3}{4}}$.

Answer:

$3^{\frac{3}{4}}$ (corresponding to the option with the expression $3^{\frac{3}{4}}$)