Simplifying the Expression (x^4y/x^9y^5)^2
This article will guide you through simplifying the expression (x^4y/x^9y^5)^2. Let's break down the steps using the rules of exponents.
Understanding the Properties of Exponents
To simplify the given expression, we need to recall the following properties of exponents:
 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
Let's simplify the expression step by step:

Apply the Power of a Power rule: (x^4y/x^9y^5)^2 = x^8y^2 / x^18y^10

Apply the Quotient of Powers rule: x^8y^2 / x^18y^10 = x^(818)y^(2(10))

Simplify the exponents: x^(818)y^(2(10)) = x^10y^8

Apply the Negative Exponent rule: x^10y^8 = 1/x^10y^8
Therefore, the simplified form of (x^4y/x^9y^5)^2 is 1/x^10y^8.
Conclusion
By applying the rules of exponents, we successfully simplified the given expression. Remember to always work through the steps systematically, using the appropriate properties to achieve the desired simplification.