Chapter 11.2, Problem 46PPE

To determine

Simplify the complex fraction: $\frac{\frac{3}{3d^2+5d-2}}{\frac{3}{2d+4}}$

Answer to Problem 46PPE

  $\frac{\frac{3}{3d^2+5d-2}}{\frac{3}{2d+4}}=\frac{2}{2d-1}$

Explanation of Solution

Given:

Simplify the complex fraction: $\frac{\frac{3}{3d^2+5d-2}}{\frac{3}{2d+4}}$

Concept Used:

Find the factors of numerator and denominator, and then cancel the common factor and simplify.

For division of rational expression, we always flip the 2^nd expression and put a multiplication sign.

Rule: $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

Calculation:

  $\begin{array}{l}\frac{\frac{3}{3d^2+5d-2}}{\frac{3}{2d+4}}=\frac{3}{3d^2+5d-2}÷\frac{3}{2d+4}\\ \\ \frac{3}{3d^2+5d-2}·\frac{2d+4}{3}=\frac{3}{(2d-1)(d+2)}·\frac{2(d+2)}{3}\\ \\ \frac{d+2}{d+2}·\frac{3}{3}·\frac{2}{(2d-1)}=\frac{2}{2d-1}\end{array}$

Thus, $\frac{\frac{3}{3d^2+5d-2}}{\frac{3}{2d+4}}=\frac{2}{2d-1}$