Let x, y be positive real numbers and m, n be positive integers, The maximum value of the expression
`(x^(m)y^(n))/((1+x^(2m))(1+y^(2n)))` is

Answer

To find the maximum value of the expression

\[ E = \frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})} \]

where \(x\) and \(y\) are positive real numbers and \(m\) and \(n\) are positive integers, we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 1: Apply AM-GM Inequality

We start by applying the AM-GM inequality to the terms in the denominators:

1. For the term \(1 + x^{2m}\):
\[ \frac{1 + x^{2m}}{2} \geq \sqrt{1 \cdot x^{2m}} = x^m \]
This implies:
\[ 1 + x^{2m} \geq 2x^m \]

2. For the term \(1 + y^{2n}\):
\[ \frac{1 + y^{2n}}{2} \geq \sqrt{1 \cdot y^{2n}} = y^n \]
This implies:
\[ 1 + y^{2n} \geq 2y^n \]

### Step 2: Combine the Inequalities

Now, we can combine these two inequalities:

\[ (1 + x^{2m})(1 + y^{2n}) \geq (2x^m)(2y^n) = 4x^m y^n \]

### Step 3: Substitute Back into the Expression

Substituting this result back into our expression for \(E\):

\[ E = \frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})} \leq \frac{x^m y^n}{4 x^m y^n} = \frac{1}{4} \]

### Step 4: Conclusion

Thus, we find that the maximum value of the expression \(E\) is:

\[ \boxed{\frac{1}{4}} \]