Simplifying Algebraic Expressions: (x^2y^4z)^5/(xy)^2
This article will guide you through the process of simplifying the algebraic expression (x^2y^4z)^5/(xy)^2. We'll use the properties of exponents to break down the expression stepbystep.
Understanding the Properties of Exponents
Before we begin, let's recall some key properties of exponents:
 Power of a Product: (ab)^n = a^n * b^n
 Power of a Power: (a^m)^n = a^(m*n)
 Division of Powers with the Same Base: a^m / a^n = a^(mn)
Simplifying the Expression

Apply the Power of a Product rule:
(x^2y^4z)^5 = x^(25) * y^(45) * z^5 = x^10 * y^20 * z^5
(xy)^2 = x^(12) * y^(12) = x^2 * y^2

Rewrite the original expression:
(x^2y^4z)^5/(xy)^2 = (x^10 * y^20 * z^5) / (x^2 * y^2)

Apply the Division of Powers with the Same Base rule:
x^10 / x^2 = x^(102) = x^8 y^20 / y^2 = y^(202) = y^18

Combine the simplified terms:
(x^10 * y^20 * z^5) / (x^2 * y^2) = x^8 * y^18 * z^5
Final Simplified Expression
Therefore, the simplified form of (x^2y^4z)^5/(xy)^2 is x^8y^18z^5.