College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter R - Section R.5 - Rational Expressions - R.5 Exercises - Page 48: 69

Answer

$\text{The simplified form of the given complex fraction is } \dfrac{m^3-4m-1}{m-2}$.

Work Step by Step

$ \begin{array}{ l l } =\dfrac{m-\dfrac{1}{( m+2)( m-2)}}{\dfrac{1}{m+2}} & \begin{array}{{l}} \mathrm{Factor} \ m^{2} -4\ \mathrm{using\ the\ difference\ of\ }\\ \mathrm{two \space squares\ property} :( m+2)( m\ -\ 2) \end{array}\\ & \\ =\dfrac{\left( m-\dfrac{1}{( m+2)( m-2)}\right) \cdot ( m+2)( m-2)}{\dfrac{1}{m+2} \cdot ( m+2)( m-2)} & \begin{array}{{l}} \mathrm{Multiply\ the\ numerator\ and\ }\\ \mathrm{denominator\ of\ the\ complex\ }\\ \mathrm{fraction\ by\ the\ LCD\ which\ }\\ \mathrm{is} \ ( m+2)( m\ -\ 2). \end{array}\\ & \\ =\dfrac{m( m+2)( m-2) -\dfrac{( m+2)( m-2)}{( m+2)( m-2)}}{\dfrac{m+2}{m+2} \cdot ( m-2)} & \begin{array}{{l}} \mathrm{Apply\ the\ distributive\ property\ in\ }\\ \mathrm{the\ numerator} .\\ \mathrm{Apply\ the\ rule} \ a\cdot \dfrac{b}{c} =\dfrac{ab}{c} \end{array}\\ & \\ =\dfrac{m( m+2)( m-2) -1}{1\cdot ( m-2)} & \mathrm{Apply\ the\ rule} \ \dfrac{a}{a} =1\\ & \\ =\dfrac{m\left( m^{2} -4\right) -1}{m-2} & \mathrm{Express} \ ( m+2)( m-2) \ \mathrm{as} \ m^{2} -4.\\ & \\ =\dfrac{m^{3} -4m-1}{m-2} & \mathrm{Apply\ the\ distributive\ property.} \end{array}$
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