Value of ` y = sqrt((a^(y/x))/(a_(x/y))) xx sqrt((a_(x/z))/(a_(y/x))) xx sqrt((a_(x/y))/(a_(x/z)))` is

Options

A. 1. `a^(xyz)`
B. 2. `a^(1/(xyz))`
C. 3. `1`
D. 4. `a`

Answer

To solve the expression \( y = \sqrt{\frac{a^{y/x}}{a^{x/y}}} \times \sqrt{\frac{a^{x/z}}{a^{y/x}}} \times \sqrt{\frac{a^{x/y}}{a^{x/z}}} \), we can follow these steps:

### Step 1: Simplify each square root
We can use the property of indices that states \( \frac{a^m}{a^n} = a^{m-n} \).

1. **First term:**
\[ \sqrt{\frac{a^{y/x}}{a^{x/y}}} = \sqrt{a^{(y/x) - (x/y)}} \]

2. **Second term:**
\[ \sqrt{\frac{a^{x/z}}{a^{y/x}}} = \sqrt{a^{(x/z) - (y/x)}} \]

3. **Third term:**
\[ \sqrt{\frac{a^{x/y}}{a^{x/z}}} = \sqrt{a^{(x/y) - (x/z)}} \]

### Step 2: Combine the square roots
Using the property of square roots, we can combine the terms:
\[ y = \sqrt{a^{(y/x) - (x/y)}} \times \sqrt{a^{(x/z) - (y/x)}} \times \sqrt{a^{(x/y) - (x/z)}} \]
This can be rewritten as:
\[ y = \sqrt{a^{\left((y/x) - (x/y) + (x/z) - (y/x) + (x/y) - (x/z)\right)}} \]

### Step 3: Simplify the exponent
Notice that the terms \( (y/x) \), \( (x/y) \), \( (x/z) \), and \( (y/x) \) will cancel out:
\[ (y/x) - (y/x) + (x/z) - (x/z) + (x/y) - (x/y) = 0 \]
Thus, we have:
\[ y = \sqrt{a^0} \]

### Step 4: Evaluate the square root
Since \( a^0 = 1 \):
\[ y = \sqrt{1} = 1 \]

### Final Answer
The value of \( y \) is \( 1 \).

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