If `x=a((m^(2)+n^(2))/(2mn))^(1//2)` , `a`, `m`, `n gt 0`, `m gt n`, find the value of `[((x^(2)+a^(2))^(1//2)+(x^(2)-a^(2))^(1//2))/((x^(2)+a^(2))^(1//2)-(x^(2)-a^(2))^(1//2))]`

Answer

To solve the problem, we start with the given expression for \( x \):

\[ x = a \left( \frac{m^2 + n^2}{2mn} \right)^{1/2} \]

We need to find the value of:

\[ \frac{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}}{\sqrt{x^2 + a^2} - \sqrt{x^2 - a^2}} \]

### Step 1: Calculate \( x^2 \)

First, we calculate \( x^2 \):

\[ x^2 = a^2 \left( \frac{m^2 + n^2}{2mn} \right) \]

### Step 2: Calculate \( x^2 + a^2 \)

Next, we find \( x^2 + a^2 \):

\[ x^2 + a^2 = a^2 \left( \frac{m^2 + n^2}{2mn} + 1 \right) = a^2 \left( \frac{m^2 + n^2 + 2mn}{2mn} \right) = a^2 \left( \frac{(m+n)^2}{2mn} \right) \]

### Step 3: Calculate \( x^2 - a^2 \)

Now, we calculate \( x^2 - a^2 \):

\[ x^2 - a^2 = a^2 \left( \frac{m^2 + n^2}{2mn} - 1 \right) = a^2 \left( \frac{m^2 + n^2 - 2mn}{2mn} \right) = a^2 \left( \frac{(m-n)^2}{2mn} \right) \]

### Step 4: Calculate \( \sqrt{x^2 + a^2} \)

Now we find \( \sqrt{x^2 + a^2} \):

\[ \sqrt{x^2 + a^2} = \sqrt{a^2 \left( \frac{(m+n)^2}{2mn} \right)} = a \frac{m+n}{\sqrt{2mn}} \]

### Step 5: Calculate \( \sqrt{x^2 - a^2} \)

Next, we find \( \sqrt{x^2 - a^2} \):

\[ \sqrt{x^2 - a^2} = \sqrt{a^2 \left( \frac{(m-n)^2}{2mn} \right)} = a \frac{m-n}{\sqrt{2mn}} \]

### Step 6: Substitute into the main expression

Now we substitute these results into the main expression:

\[ \frac{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}}{\sqrt{x^2 + a^2} - \sqrt{x^2 - a^2}} = \frac{a \frac{m+n}{\sqrt{2mn}} + a \frac{m-n}{\sqrt{2mn}}}{a \frac{m+n}{\sqrt{2mn}} - a \frac{m-n}{\sqrt{2mn}}} \]

### Step 7: Simplify the expression

This simplifies to:

\[ = \frac{a \left( \frac{(m+n) + (m-n)}{\sqrt{2mn}} \right)}{a \left( \frac{(m+n) - (m-n)}{\sqrt{2mn}} \right)} = \frac{a \frac{2m}{\sqrt{2mn}}}{a \frac{2n}{\sqrt{2mn}}} = \frac{m}{n} \]

### Final Answer

Thus, the value of the given expression is:

\[ \frac{m}{n} \]