Find the value of ` (4/(log_(2)(2sqrt3))+2/(log_(3) (2sqrt3)))^(2)`.

Explanation

To solve the expression \( \left( \frac{4}{\log_2(2\sqrt{3})} + \frac{2}{\log_3(2\sqrt{3})} \right)^2 \), we will follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula.
Using the property of logarithms, we can express \( \log_a(b) \) as \( \frac{\log(b)}{\log(a)} \). Therefore, we have:

\[ \log_2(2\sqrt{3}) = \frac{\log(2\sqrt{3})}{\log(2)} \quad \text{and} \quad \log_3(2\sqrt{3}) = \frac{\log(2\sqrt{3})}{\log(3)} \]

### Step 2: Substitute these into the expression.
Now substituting these into our expression gives:

\[ \frac{4}{\log_2(2\sqrt{3})} = \frac{4 \log(2)}{\log(2\sqrt{3})} \quad \text{and} \quad \frac{2}{\log_3(2\sqrt{3})} = \frac{2 \log(3)}{\log(2\sqrt{3})} \]

So the expression becomes:

\[ \left( \frac{4 \log(2)}{\log(2\sqrt{3})} + \frac{2 \log(3)}{\log(2\sqrt{3})} \right)^2 \]

### Step 3: Combine the fractions.
We can combine the terms in the parentheses:

\[ = \left( \frac{4 \log(2) + 2 \log(3)}{\log(2\sqrt{3})} \right)^2 \]

### Step 4: Factor out common terms.
Notice that we can factor out a 2 from the numerator:

\[ = \left( \frac{2(2 \log(2) + \log(3))}{\log(2\sqrt{3})} \right)^2 \]

### Step 5: Simplify the logarithm in the denominator.
Now, we simplify \( \log(2\sqrt{3}) \):

\[ \log(2\sqrt{3}) = \log(2) + \log(\sqrt{3}) = \log(2) + \frac{1}{2} \log(3) \]

### Step 6: Substitute back into the expression.
Now substituting this back gives:

\[ = \left( \frac{2(2 \log(2) + \log(3))}{\log(2) + \frac{1}{2} \log(3)} \right)^2 \]

### Step 7: Evaluate the expression.
Let \( x = \log(2) \) and \( y = \log(3) \). Then we have:

\[ = \left( \frac{2(2x + y)}{x + \frac{1}{2}y} \right)^2 \]

### Step 8: Simplify further.
Now, let's simplify the fraction:

\[ = \left( \frac{2(2x + y)}{x + \frac{1}{2}y} \right)^2 = \left( \frac{4x + 2y}{x + \frac{1}{2}y} \right)^2 \]

### Step 9: Final simplification.
Now we can evaluate this expression. The numerator is \( 4x + 2y \) and the denominator is \( x + \frac{1}{2}y \).

### Final Answer:
After evaluating, we find that:

\[ = 16 \]

Thus, the final value of the expression is \( 16 \).

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