`(root5root5(a^(5)))^(10)` is equal to

Options

A. `a^(2)`
B. 1
C. `a^(1//5)`
D. `a^(5)`

Answer

To solve the expression \((\sqrt[5]{\sqrt[5]{a^5}})^{10}\), we will break it down step by step.

### Step 1: Understand the Expression
The expression can be rewritten as:
\[ \left(\sqrt[5]{\sqrt[5]{a^5}}\right)^{10} \]
This means we have a double fifth root of \(a^5\).

### Step 2: Simplify the Inner Root
First, let's simplify the inner fifth root:
\[ \sqrt[5]{a^5} = a^{5 \cdot \frac{1}{5}} = a^1 = a \]
So, we can rewrite the expression as:
\[ \left(\sqrt[5]{a}\right)^{10} \]

### Step 3: Apply the Outer Root
Now, we need to simplify the outer fifth root:
\[ \sqrt[5]{a} = a^{\frac{1}{5}} \]
Thus, we can rewrite the expression as:
\[ \left(a^{\frac{1}{5}}\right)^{10} \]

### Step 4: Apply the Power Rule
Using the power of a power rule \((x^m)^n = x^{m \cdot n}\), we can simplify further:
\[ \left(a^{\frac{1}{5}}\right)^{10} = a^{\frac{1}{5} \cdot 10} = a^{2} \]

### Final Answer
The final simplified expression is:
\[ \boxed{a^2} \]

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