Simplified form of $[(\sqrt[5]{x^{-\frac{3}{5}}})^{-\frac{5}{3}}]^5$ is

### Concept & Formula This problem tests the chaining of power rules and radical rules. Instead of working layer by layer, you can convert all radicals into fractional exponents and multiply the entire chain of exponents together. $$ \sqrt[n]{x} = x^{\frac{1}{n}} $$ $$ (((x^a)^b)^c)^d = x^{a \times b \times c \times d} $$ ### Step-by-Step Solution * **Given:** The nested expression is $[(\sqrt[5]{x^{-\frac{3}{5}}})^{-\frac{5}{3}}]^5$. * **Calculation:** 1. First, convert the 5th root into a fractional exponent to standardize all operations. The 5th root becomes an exponent of $\frac{1}{5}$. 2. Rewrite the entire expression using only exponents: $(((x^{-\frac{3}{5}})^{\frac{1}{5}})^{-\frac{5}{3}})^5$ 3. Since these are all nested powers, apply the power of a power rule by multiplying the entire chain of exponents together: Exponent $= -\frac{3}{5} \times \frac{1}{5} \times -\frac{5}{3} \times 5$ 4. Notice the negative signs cancel out (negative $\times$ negative = positive): Exponent $= \frac{3}{5} \times \frac{1}{5} \times \frac{5}{3} \times 5$ 5. Simplify the fractions. The $3$s cancel, and the two $5$s in the numerator cancel the two $5$s in the denominator: Exponent $= 1$ 6. The final simplified base and exponent is: $x^1 = x$ ### Exam Strategy & Shortcut Avoid rewriting the variable $x$ over and over. Just extract the exponents and multiply them linearly. You have $-3/5$. It is under a 5th root, so multiply by $1/5$. It is raised to $-5/3$, so multiply by that. Finally, it is raised to $5$, so multiply by that. $-\frac{3}{5} \times \frac{1}{5} \times -\frac{5}{3} \times 5$. Visually cross out the corresponding numerators and denominators. Everything cancels out perfectly leaving a $1$. The answer is just $x$. ### Common Pitfall Students often get confused by the sequence and try to evaluate the innermost terms fully before moving outward, which leads to messy fractions like $x^{-3/25}$ and increases the likelihood of a calculation error. Stringing them together as a single multiplication problem is much safer. ### Final Answer **Therefore, the correct answer is $x$.**