Simplifying Exponential Expressions: A StepbyStep Guide
This article will guide you through the process of simplifying the expression (x^3y^2)^4/(y^6x^4)^2. We'll break down the problem into manageable steps using the properties of exponents.
Understanding the Properties of Exponents
Before diving into the simplification, let's review some key exponent rules that will be used:
 Product of Powers: x^m * x^n = x^(m+n)
 Quotient of Powers: x^m / x^n = x^(mn)
 Power of a Power: (x^m)^n = x^(m*n)
 Negative Exponent: x^n = 1/x^n
Simplifying the Expression

Distribute the outer exponents:
 Applying the "Power of a Power" rule, we get: (x^(3*4)y^(2*4)) / (y^(6*2)x^(4*2))
 This simplifies to: (x^12 y^8) / (y^12 x^8)

Move terms with negative exponents:
 Utilizing the "Negative Exponent" rule, we can rewrite the expression as: (x^12 * 1/y^8) / (1/y^12 * x^8)

Combine terms with the same base:
 Applying the "Quotient of Powers" rule: x^(128) * (y^12/y^8)
 This simplifies to: x^4 * y^4
Final Result
The simplified expression is x^4y^4.
By applying the fundamental properties of exponents, we successfully simplified the complex expression into a more manageable form. Remember to practice these rules to master simplifying expressions with exponents.