Simplifying (x^2/5 y^4/10)^5
This problem involves simplifying an expression with negative exponents and fractions. Let's break down the steps:
Understanding the Rules
 Negative Exponents: A term raised to a negative exponent is equal to its reciprocal raised to the positive version of that exponent.
 Example: x^2 = 1/x^2
 Fractional Exponents: When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent.
 Example: (a/b)^n = a^n/b^n
 Power of a Power: When a power is raised to another power, the exponents are multiplied.
 Example: (x^m)^n = x^(m*n)
Applying the Rules

Reciprocal: Since the expression is raised to a negative exponent, we take its reciprocal and change the exponent to positive:
 (x^2/5 y^4/10)^5 = (1 / (x^2/5 y^4/10))^5

Simplifying the Reciprocal:
 (1 / (x^2/5 y^4/10))^5 = (5y^4/10 / x^2)^5

Power of a Power: Apply the power of a power rule to both the numerator and the denominator:
 (5y^4/10 / x^2)^5 = (5^5 * y^(45) / 10^5) / x^(25)

Simplifying Exponents: Calculate the exponents:
 (5^5 * y^(45) / 10^5) / x^(25) = (3125 y^20 / 100000) / x^10

Final Simplification: Combine the constants:
 (3125 y^20 / 100000) / x^10 = (125 y^20) / (400 x^10)
The Result
The simplified form of (x^2/5 y^4/10)^5 is (125 y^20) / (400 x^10).