Problem 64 - College Algebra

Question

Simplify each complex rational expression. $$\frac{1-\frac{1}{x}}{x y}$$

Step-by-Step Solution

Answer

The simplified fractional expression is \(\frac{1 - x^{-1}}{y^2}\)

Step 1: Identify Inner Fraction

Firstly, identify the inner fraction, which is \(\frac{1}{x}\) in the numerator. This will be the main focus for simplification.

Step 2: Simplify Inner Fraction

Change the form of the inner fraction to help in further simplification. Rewrite \(\frac{1}{x}\) as \(x^{-1}\). This simplifies the numerator to \(1 - x^{-1}\).

Step 3: Simplify Outer Fraction

Now, the main fraction looks like \(\frac{1 - x^{-1}}{x y}\). The numerator \(1 - x^{-1}\) is equivalent to \((x - 1) / x\). Therefore, the whole expression simplifies to \(\frac{(x - 1)}{xy^2}\).

Step 4: Final Simplification

To get our final answer, we can divide both numerator and denominator by \(x\) which will lead us with \(\frac{1 - x^{-1}}{y^2}\).