To determine which exponential expression is equivalent to [tex]\((\sqrt[3]{t})^2\)[/tex], we need to simplify it step-by-step.
1. Rewrite the Radicals as Exponents:
The expression [tex]\(\sqrt[3]{t}\)[/tex] can be rewritten in exponential form. By definition, the cube root of [tex]\(t\)[/tex] is [tex]\(t\)[/tex] raised to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\sqrt[3]{t} = t^{\frac{1}{3}}
\][/tex]
2. Apply the Power Rule for Exponents:
We need to raise this expression to the power of 2:
[tex]\[
(\sqrt[3]{t})^2 = \left(t^{\frac{1}{3}}\right)^2
\][/tex]
3. Simplify the Exponent:
When raising a power to another power, you multiply the exponents. Therefore, we multiply [tex]\(\frac{1}{3}\)[/tex] by 2:
[tex]\[
\left(t^{\frac{1}{3}}\right)^2 = t^{\frac{1}{3} \cdot 2} = t^{\frac{2}{3}}
\][/tex]
The equivalent expression is [tex]\(t^{\frac{2}{3}}\)[/tex], which corresponds to option (B).
Thus, the correct answer is:
[tex]\[
\boxed{t^{\frac{2}{3}}}
\][/tex]
Therefore, option (B) is the correct choice.
