%2F(x-*-y-%5E-3))-%5E---2-*-((x-%5E-7-*-y-%5E---3)%2F(x-%5E---4-*-y-%5E-6))-%5E-(--1%2F2)-x-ne0-y-ne0.jpeg "((x ^ - 5 * Y ^ 4)/(x * Y ^ 3)) ^ - 2 * ((x ^ 7 * Y ^ - 3)/(x ^ - 4 * Y ^ 6)) ^ (- 1/2) X Ne0 Y Ne0")
Simplifying Rational Expressions: A Step-by-Step Guide
In this article, we will simplify a complex rational expression involving exponential functions. The expression is:
$\left(\frac{x^{-5}y^4}{xy^3}\right)^{-2}\left(\frac{x^7y^{-3}}{x^{-4}y^6}\right)^{-\frac{1}{2}}$
where $x\neq 0$ and $y\neq 0$.
Step 1: Simplify the Expressions Inside the Parentheses
Let's start by simplifying the expressions inside the parentheses.
$\frac{x^{-5}y^4}{xy^3} = \frac{y^4}{x^6y^3} = x^{-6}y$
$\frac{x^7y^{-3}}{x^{-4}y^6} = x^{11}y^{-9}$
Step 2: Apply the Exponent Rules
Now, let's apply the exponent rules to simplify the expression.
$\left(\frac{y^4}{x^6y^3}\right)^{-2} = \left(x^6y^3\right)^2 = x^{12}y^6$
$\left(x^{11}y^{-9}\right)^{-\frac{1}{2}} = x^{-\frac{11}{2}}y^\frac{9}{2}$
Step 3: Multiply the Results
Finally, let's multiply the results to get the final simplified expression.
$x^{12}y^6 \cdot x^{-\frac{11}{2}}y^\frac{9}{2} = x^\frac{13}{2}y^\frac{33}{2}$
Therefore, the simplified expression is:
$\boxed{x^\frac{13}{2}y^\frac{33}{2}}$
