The problem asks which expression is equivalent to $(\sqrt[3]{x})^6$. Let's solve it step by step:

Step 1: Rewrite the square root as a fractional exponent.

The cube root of $x$, denoted as $\sqrt[3]{x}$, can be written as: $x^{\frac{1}{3}}$

Step 2: Apply the exponentiation rule.

When raising a power to another power, multiply the exponents: $\left(x^{\frac{1}{3}}\right)^6 = x^{\frac{1}{3} \cdot 6}$

Step 3: Simplify the exponent.

$\frac{1}{3} \cdot 6 = 2$

So, $(\sqrt[3]{x})^6 = x^2$.

Final Answer:

The expression equivalent to $(\sqrt[3]{x})^6$ is $x^2$.

Do you want more details about exponentiation? Here are five related questions to explore this further:

1. What are the rules for manipulating fractional exponents?
2. How do cube roots and square roots differ when written as exponents?
3. What happens when the base of the exponent is negative?
4. Can this method be extended to expressions like $(\sqrt[n]{x^m})^p$?
5. How can you simplify $(x^{1/2})^4$?

Tip:

When raising a root to a power, always convert the root to a fractional exponent first—it simplifies the calculations!