Simplifying Algebraic Expressions: (a^2b^3)^2/(ab)^2
This article will guide you through the process of simplifying the algebraic expression (a^2b^3)^2/(ab)^2. We will break down the steps and explain the rules involved.
Understanding the Rules
Before we start simplifying, let's recall some key rules of exponents:
 Power of a power: (x^m)^n = x^(m*n)
 Negative exponent: x^n = 1/x^n
 Division of powers: x^m / x^n = x^(mn)
Simplifying the Expression

Apply the power of a power rule to both numerator and denominator:
 (a^2b^3)^2 = a^(22) * b^(32) = a^4b^6
 (ab)^2 = a^(21) * b^(21) = a^2b^2

Apply the negative exponent rule to the denominator:
 a^2b^2 = 1/(a^2b^2)

Rewrite the expression with the simplified terms:
 (a^2b^3)^2 / (ab)^2 = (a^4b^6) / (1/(a^2b^2))

Dividing by a fraction is the same as multiplying by its reciprocal:
 (a^4b^6) / (1/(a^2b^2)) = (a^4b^6) * (a^2b^2/1)

Multiply the terms in the numerator:
 (a^4b^6) * (a^2b^2) = a^(4+2) * b^(6+2) = a^6b^8
Final Simplified Expression
Therefore, the simplified form of the expression (a^2b^3)^2/(ab)^2 is a^6b^8.