Chapter A.3, Problem 68E

Solution

Simplify the compound fraction.

  $\frac{2}{(x+h+2)(x+2)}$ .

Given:

The given compound fraction is $\frac{\frac{x+h}{x+h+2}-\frac{x}{x+2}}{h}$ .

Concept Used:

• Compound fraction is a fraction with fractions in the numerator or denominator.
• First multiply the denominator of the fraction by the whole number. 
• Then, add that number to the numerator of the fraction, and keep the original denominator the same.

Calculation:

The given compound fraction is $\frac{\frac{x+h}{x+h+2}-\frac{x}{x+2}}{h}$ .

  $\begin{array}{c}\frac{\frac{x+h}{x+h+2}-\frac{x}{x+2}}{h}=\frac{\frac{(x+h)(x+2)-x(x+h+2)}{(x+h+2)(x+2)}}{h}\\ =\frac{\frac{(x^2+2x+hx+2h)-(x^2+xh+2x)}{(x+h+2)(x+2)}}{h}\\ =\frac{x^2+2x+hx+2h-x^2-xh-2x}{h(x+h+2)(x+2)}\\ =\frac{2h}{h(x+h+2)(x+2)}\\ =\frac{2}{(x+h+2)(x+2)}\end{array}$

Thus by simplifying the compound fraction $\frac{\frac{x+h}{x+h+2}-\frac{x}{x+2}}{h}$ the solution is $\frac{2}{(x+h+2)(x+2)}$