Simplifying Expressions with Exponents
This article will guide you through the simplification of the expression (2xy^2)^4(2x^3y^4)^2.
Understanding the Rules
Before we dive into the simplification, let's review some key rules of exponents:
 Product of powers: x^m * x^n = x^(m+n)
 Power of a product: (xy)^n = x^n * y^n
 Power of a power: (x^m)^n = x^(m*n)
Applying the Rules

Distribute the exponents:
 (2xy^2)^4 = (2)^4 * x^4 * (y^2)^4 = 16x^4y^8
 (2x^3y^4)^2 = 2^2 * (x^3)^2 * (y^4)^2 = 4x^6y^8

Multiply the resulting terms:
 16x^4y^8 * 4x^6y^8 = 64x^(4+6)y^(8+8) = 64x^10y^16
The Final Answer
Therefore, the simplified form of (2xy^2)^4(2x^3y^4)^2 is 64x^10y^16.
Conclusion
By applying the rules of exponents, we successfully simplified the complex expression. Remember to always break down the problem into smaller steps and use the appropriate rules for each step. This approach ensures accuracy and helps you avoid common errors.