Simplify : `[root(3)root(6)(5^(9))]^(4)[root(3)root(6)(5^(9)]]^(4)`

Options

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

Answer

To simplify the expression \([ \sqrt[3]{\sqrt{6}(5^{9})}]^{4}[\sqrt[3]{\sqrt{6}(5^{9})}]^{4}\), we can follow these steps:

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

### Step 2: Combine the powers
Since both parts of the expression are the same, we can combine them:
\[ \left(\sqrt[3]{\sqrt{6} \cdot 5^{9}}\right)^{4 + 4} = \left(\sqrt[3]{\sqrt{6} \cdot 5^{9}}\right)^{8} \]

### Step 3: Simplify the cube root
Next, we simplify the cube root:
\[ \sqrt[3]{\sqrt{6} \cdot 5^{9}} = (\sqrt{6})^{\frac{1}{3}} \cdot (5^{9})^{\frac{1}{3}} = (6^{\frac{1}{2}})^{\frac{1}{3}} \cdot 5^{3} = 6^{\frac{1}{6}} \cdot 5^{3} \]

### Step 4: Substitute back into the expression
Now, substitute this back into the expression:
\[ \left(6^{\frac{1}{6}} \cdot 5^{3}\right)^{8} \]

### Step 5: Apply the power of a product rule
Using the power of a product rule, we can distribute the exponent:
\[ (6^{\frac{1}{6}})^{8} \cdot (5^{3})^{8} = 6^{\frac{8}{6}} \cdot 5^{24} \]

### Step 6: Simplify the exponent of 6
Now simplify \(6^{\frac{8}{6}}\):
\[ 6^{\frac{8}{6}} = 6^{\frac{4}{3}} \]

### Step 7: Final expression
Combining everything, we get:
\[ 6^{\frac{4}{3}} \cdot 5^{24} \]

### Conclusion
Thus, the simplified form of the given expression is:
\[ 6^{\frac{4}{3}} \cdot 5^{24} \]