Simplified form of `[(root(5)(x^(-3//5)))^(-5//3)]^(5)` is

Options

A. `x^(5)`
B. `x^(-5)`
C. `x`
D. `(1)/(x)`

Answer

To simplify the expression \(\left[\sqrt[5]{x^{-\frac{3}{5}}}\right]^{-\frac{5}{3}}^5\), we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten using exponent rules:
\[ \left[\sqrt[5]{x^{-\frac{3}{5}}}\right]^{-\frac{5}{3}} = \left[x^{-\frac{3}{5}}^{\frac{1}{5}}\right]^{-\frac{5}{3}} \]
This means we can express the 5th root as an exponent of \(\frac{1}{5}\).

### Step 2: Simplify the exponent
Now, we simplify the exponent:
\[ x^{-\frac{3}{5} \cdot \frac{1}{5}} = x^{-\frac{3}{25}} \]
Thus, we have:
\[ \left[x^{-\frac{3}{25}}\right]^{-\frac{5}{3}} \]

### Step 3: Apply the exponent rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we simplify further:
\[ x^{-\frac{3}{25} \cdot -\frac{5}{3}} = x^{\frac{3 \cdot 5}{25 \cdot 3}} = x^{\frac{15}{75}} = x^{\frac{1}{5}} \]

### Step 4: Raise to the power of 5
Now we raise this result to the power of 5:
\[ \left[x^{\frac{1}{5}}\right]^5 = x^{\frac{1}{5} \cdot 5} = x^1 = x \]

### Final Result
Thus, the simplified form of the original expression is:
\[ \boxed{x} \]