Problem 71 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}$$

Step-by-Step Solution

Answer

-2x

Step 1: Get the same denominator

Firstly, the fractions need to be written under the same denominator, which will allow for simplification. Algebraically, this is achieved by expressing the numerator as \(x^{2}(x+h)^{2}\), giving \(\frac{x^{2}-(x+h)^{2}}{h(x^{2}(x+h)^{2})}\).

Step 2: Simplifying the numerator

The numerator can be simplified via expansion and combining like terms. The result is \(\frac{-2xh-h^{2}}{h(x^{2}(x+h)^{2})}\).

Step 3: Cancel the complex fraction

Cancelled term \(h\) between numerator and denominator gives \(-2x-\frac{h}{x^{2}(x+h)^{2}}\).

Step 4: Final simplification

Now, considering \(h\) approaches 0, the final ans becomes \(-2x\).