Problem 73 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2} y-x^{2}}{x^{3}-x^{3} y}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression \(\frac{x^{2} y-x^{2}}{x^{3}-x^{3} y}\) is \(-1/x.\)

Step 1: Factor out common terms in the numerator

Notice that \(x^2\) is a common term in the numerator. Factoring it out gives, \(x^{2}(y - 1).\)

Step 2: Factor out common terms in the denominator

In the denominator, \(x^3\) is a common term. Factoring this out gives, \(x^{3}(1 - y).\)

Step 3: Reduce the rational expression

The simplified form of the rational expression by canceling out \(y - 1\) in the numerator and \(1 - y\) in the denominator (remember, \(y - 1 = -(1 - y)\)) is \(-x^2 / x^3 = -1/x\).