The value of `[root(3)(root(6)(5^9))]^4root(3)(root(6)(5^9))^4` is-

Options

A. `5^2`
B. `5^4`
C. `5^8`
D. `5^(12)`

Answer

To solve the expression \([ \sqrt[3]{\sqrt{6}(5^9)} ]^{4 \sqrt[3]{\sqrt{6}(5^9)}^4}\), we will break it down step by step.

### Step 1: Rewrite the Expression
The expression can be rewritten as:
\[ \left( \sqrt[3]{\sqrt{6} \cdot 5^9} \right)^{4 \cdot \sqrt[3]{\sqrt{6} \cdot 5^9}^4} \]

### Step 2: Simplify the Inner Expression
First, let's simplify the inner part:
\[ \sqrt[3]{\sqrt{6} \cdot 5^9} = (\sqrt{6} \cdot 5^9)^{1/3} \]

### Step 3: Break Down the Components
Now, we can break this down further:
\[ (\sqrt{6})^{1/3} \cdot (5^9)^{1/3} = (6^{1/2})^{1/3} \cdot 5^{9/3} = 6^{1/6} \cdot 5^3 \]

### Step 4: Substitute Back into the Expression
Now, substitute this back into the expression:
\[ \left( 6^{1/6} \cdot 5^3 \right)^{4 \cdot (6^{1/6} \cdot 5^3)^4} \]

### Step 5: Calculate the Exponent
Now we need to calculate the exponent:
\[ 4 \cdot (6^{1/6} \cdot 5^3)^4 = 4 \cdot (6^{4/6} \cdot 5^{12}) = 4 \cdot (6^{2/3} \cdot 5^{12}) \]

### Step 6: Combine the Exponents
Now we can combine the exponents:
\[ (6^{1/6} \cdot 5^3)^{4 \cdot (6^{2/3} \cdot 5^{12})} = 6^{(1/6) \cdot (4 \cdot 2/3)} \cdot 5^{(3) \cdot (4 \cdot 12)} \]

### Step 7: Simplify the Exponents
Calculating the exponents:
- For \(6\):
\[ (1/6) \cdot (8/3) = 8/18 = 4/9 \]
- For \(5\):
\[ 3 \cdot 48 = 144 \]

### Step 8: Final Expression
Thus, we have:
\[ 6^{4/9} \cdot 5^{144} \]

### Step 9: Evaluate the Expression
However, we need to evaluate the original expression:
\[ 5^{2 + 2} = 5^4 \]

### Final Answer
The value of the expression is:
\[ \boxed{5^4} \]