Question

choose all the expressions that are equivalent to: $(3x^{5})^{\frac{3}{5}}$
$3sqrt5{x^{5}}$
$3sqrt5{x^{3}}$
$sqrt5{3x^{3}}$
$xsqrt5{3x^{2}}$
$sqrt5{3x^{5}}$

Explanation:

Step1: Rewrite exponent as root

Recall that $(a^m)^n=a^{mn}$ and $a^{\frac{1}{k}}=\sqrt[k]{a}$. For $(3x^5)^{\frac{1}{3}}$, this becomes $\sqrt[3]{3x^5}$.

Step2: Split the radical term

Use $\sqrt[3]{ab}=\sqrt[3]{a}\sqrt[3]{b}$ and $\sqrt[3]{x^5}=\sqrt[3]{x^3 \cdot x^2}=x\sqrt[3]{x^2}$. So:
$\sqrt[3]{3x^5}=x\sqrt[3]{3x^2}$

Step3: Verify other options

• $3\sqrt[3]{x^5}$: Incorrect, since the 3 is inside the original parenthesis, not outside.
• $3\sqrt[3]{x^3}$: Incorrect, misapplies exponent rules and constants.
• $\sqrt[3]{3x^3}$: Incorrect, uses wrong exponent for $x$.

Answer:

$\sqrt[3]{3x^5}$, $x\sqrt[3]{3x^2}$