Question P.6.11: Simplifying a Complex Rational Expression Simplify: 1/x + h ...
Simplifying a Complex Rational Expression
Simplify: \frac{\frac{1}{x+ h} – \frac{1}{x}}{h}.
We will use the method of multiplying each of the three terms, \frac{1}{x+ h}, \frac{1}{x}, and h, by the least common denominator. The least common denominator is x(x + h).
\frac{\frac{1}{x+ h} – \frac{1}{x}}{h}=\frac{(\frac{1}{x+ h} – \frac{1}{x})x(x + h)}{h x(x + h)}
Multiply the numerator and denominator by x(x + h), h ≠ 0, x ≠ 0, x ≠ -h.
=\frac{\frac{1}{x+ h}·x(x + h) – \frac{1}{x}·x(x + h)}{hx(x + h)}
Use the distributive property in the numerator.
=\frac{x – (x + h)}{hx(x + h)}
Simplify: \frac{1}{\cancel{x+h}}·x(\cancel{x+h})=x and \frac{1}{\cancel{x}}·\cancel{x}(x+h)= x + h.
=\frac{x – x – h}{hx(x + h)}
Subtract in the numerator. Remove parentheses and change the sign of each term in parentheses.
=\frac{-\overset{1}{\cancel{h}}}{\underset{1}{\cancel{h}}x(x+h)} Simplify: x – x – h = -h.
=-\frac{1}{x(x + h)}, h ≠ 0, x ≠ 0, x ≠ -h
Divide the numerator and denominator by h.