Problem 71 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified complex rational expression is \( \frac{-2x - h}{x^{2}(x+h)^{2}} \)

Step 1: Rationalize the Numerator

Content for Step 1: Begin by first rationalizing the expressions in the numerator i.e., eliminating the denominators. To do this, multiply each fraction by the denominator of the other. This gives us \( \frac{x^{2} - (x+h)^{2}}{hx^{2}(x+h)^{2}} \)

Step 2: Simplify the Numerator

Content for Step 2: Next, simplify the numerator using the difference of squares formula which gives us \( \frac{-2xh - h^{2}}{hx^{2}(x+h)^{2}} \)

Step 3: Simplify the entire fraction

Content for Step 3: Finally, we now simplify the entire fraction by taking 'h' common from the numerator and simplify it by cancelling one 'h' term in the numerator and denominator which gives us \( \frac{-2x - h}{x^{2}(x+h)^{2}} \)