Simplify `[root(3)(root(6)(2^(9)))]^(4)xx[root(6)(root(3)(2^(9)))]^(4)`.

Options

A. `2^(4)`
B. `2^(9)`
C. `2^(3)`
D. `2^(16)`

Answer

Let's simplify the expression \([ \sqrt[3]{\sqrt{6}(2^9)} ]^4 \times [ \sqrt{6}(\sqrt[3]{2^9}) ]^4\) step by step.

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ \left( \sqrt[3]{\sqrt{6} \cdot 2^9} \right)^4 \times \left( \sqrt{6} \cdot \sqrt[3]{2^9} \right)^4 \]

### Step 2: Apply the power of a product rule
Using the property \((ab)^n = a^n \cdot b^n\), we can separate the components:
\[ = \left( \sqrt[3]{\sqrt{6}} \cdot \sqrt[3]{2^9} \right)^4 \times \left( \sqrt{6} \cdot \sqrt[3]{2^9} \right)^4 \]

### Step 3: Simplify each component
Now, we simplify each part:
1. For \(\sqrt[3]{\sqrt{6}}\):
\[ \sqrt[3]{\sqrt{6}} = \sqrt[3]{6^{1/2}} = 6^{1/6} \]
2. For \(\sqrt[3]{2^9}\):
\[ \sqrt[3]{2^9} = 2^{9/3} = 2^3 \]
3. For \(\sqrt{6}\):
\[ \sqrt{6} = 6^{1/2} \]

### Step 4: Substitute back into the expression
Substituting these back into the expression gives:
\[ = \left( 6^{1/6} \cdot 2^3 \right)^4 \times \left( 6^{1/2} \cdot 2^3 \right)^4 \]

### Step 5: Apply the power of a product rule again
Now we can apply the power of a product rule:
\[ = (6^{4/6} \cdot 2^{12}) \times (6^{4/2} \cdot 2^{12}) \]

### Step 6: Simplify the powers
Calculating the powers:
1. \(6^{4/6} = 6^{2/3}\)
2. \(6^{4/2} = 6^2\)

Now we have:
\[ = (6^{2/3} \cdot 2^{12}) \times (6^2 \cdot 2^{12}) \]

### Step 7: Combine like terms
Combining the powers of \(6\) and \(2\):
\[ = 6^{2/3 + 2} \cdot 2^{12 + 12} \]
Calculating the exponents:
\[ = 6^{2/3 + 6/3} \cdot 2^{24} = 6^{8/3} \cdot 2^{24} \]

### Step 8: Final simplification
Now we can express \(6^{8/3}\) as:
\[ = (6^8)^{1/3} \cdot 2^{24} \]

### Conclusion
The final simplified expression is:
\[ = 6^{8/3} \cdot 2^{24} \]