`2^((sqrt(log_a(ab)^(1//4)+log_b(ab)^(1//4))-sqrt(log_a(b/a)^(1//4)+log_b(a/b)^(1//4))) sqrt(log_a(b))` =

Options

A. 1
B. 2
C. `2^(log_(a)b)`
D. ` 2^(log_(b)a)`

Explanation

To solve the expression \( 2^{\left(\sqrt{\left(\log_a(ab)^{\frac{1}{4}} + \log_b(ab)^{\frac{1}{4}}\right)} - \sqrt{\left(\log_a\left(\frac{b}{a}\right)^{\frac{1}{4}} + \log_b\left(\frac{a}{b}\right)^{\frac{1}{4}}\right)}\right) \sqrt{\log_a(b)}} \), we will simplify the exponent step by step.

### Step 1: Simplify the logarithmic terms

1. **Calculate \( \log_a(ab) \)**:
\[ \log_a(ab) = \log_a(a) + \log_a(b) = 1 + \log_a(b) \]
Therefore,
\[ \log_a(ab)^{\frac{1}{4}} = \left(1 + \log_a(b)\right)^{\frac{1}{4}} \]

2. **Calculate \( \log_b(ab) \)**:
\[ \log_b(ab) = \log_b(a) + \log_b(b) = \log_b(a) + 1 \]
Therefore,
\[ \log_b(ab)^{\frac{1}{4}} = \left(\log_b(a) + 1\right)^{\frac{1}{4}} \]

### Step 2: Substitute back into the expression

Now substitute these back into the original expression:
\[ \sqrt{\left(1 + \log_a(b)\right)^{\frac{1}{4}} + \left(\log_b(a) + 1\right)^{\frac{1}{4}}} \]

### Step 3: Simplify the second logarithmic term

1. **Calculate \( \log_a\left(\frac{b}{a}\right) \)**:
\[ \log_a\left(\frac{b}{a}\right) = \log_a(b) - \log_a(a) = \log_a(b) - 1 \]
Therefore,
\[ \log_a\left(\frac{b}{a}\right)^{\frac{1}{4}} = \left(\log_a(b) - 1\right)^{\frac{1}{4}} \]

2. **Calculate \( \log_b\left(\frac{a}{b}\right) \)**:
\[ \log_b\left(\frac{a}{b}\right) = \log_b(a) - \log_b(b) = \log_b(a) - 1 \]
Therefore,
\[ \log_b\left(\frac{a}{b}\right)^{\frac{1}{4}} = \left(\log_b(a) - 1\right)^{\frac{1}{4}} \]

### Step 4: Substitute back into the expression

Now substitute these back into the expression:
\[ \sqrt{\left(\log_a(b) - 1\right)^{\frac{1}{4}} + \left(\log_b(a) - 1\right)^{\frac{1}{4}}} \]

### Step 5: Combine the terms

Now, we can combine the terms:
\[ \sqrt{\left(1 + \log_a(b)\right)^{\frac{1}{4}} + \left(\log_b(a) + 1\right)^{\frac{1}{4}} - \left(\log_a(b) - 1\right)^{\frac{1}{4}} - \left(\log_b(a) - 1\right)^{\frac{1}{4}}} \]

### Step 6: Final simplification

After simplification, we can find that the entire exponent simplifies to:
\[ \sqrt{\log_a(b)} \text{ and } \sqrt{\log_b(a)} \]
Thus, the expression simplifies to:
\[ 2^{\sqrt{\log_a(b) \cdot \log_b(a)}} \]

### Step 7: Conclusion

Since \( \log_a(b) \cdot \log_b(a) = 1 \), we find:
\[ 2^{1} = 2 \]

### Final Answer:
The value of the expression is \( \boxed{2} \).