1. **State the problem:** Simplify the expression $$\frac{(32x^2 + y^2)^2}{(6.25x^4 + y^4)^3}$$. 2. **Rewrite constants as fractions or powers:** Note that $$6.25 = \left(2.5\right)^2 = \left(\frac{5}{2}\right)^2$$. 3. **Express the denominator terms:** $$6.25x^4 = \left(\frac{5}{2}\right)^2 x^4 = \left(\frac{5}{2} x^2\right)^2$$ and $$y^4 = (y^2)^2$$. 4. **Rewrite denominator as sum of squares:** $$6.25x^4 + y^4 = \left(\frac{5}{2} x^2\right)^2 + (y^2)^2$$. 5. **Check numerator:** $$32x^2 + y^2$$ cannot be factored easily with denominator terms. 6. **Rewrite numerator and denominator powers:** $$\frac{(32x^2 + y^2)^2}{\left(6.25x^4 + y^4\right)^3} = \frac{(32x^2 + y^2)^2}{\left(\left(\frac{5}{2} x^2\right)^2 + (y^2)^2\right)^3}$$. 7. **No common factors or simplifications:** Since numerator is a square of a binomial and denominator is a cube of sum of squares, no direct simplification is possible. **Final simplified form:** $$\boxed{\frac{(32x^2 + y^2)^2}{(6.25x^4 + y^4)^3}}$$