Section 1.6 : Rational Expressions
12. Perform the indicated operation in the following expression.
\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}}\]Show All Steps Hide All Steps
Start SolutionWe first need the least common denominator for this rational expression.
\[{\mbox{lcd : }}{\left( {3x + 8} \right)^3}\]Remember that we only take the highest power on each term in the denominator when setting up the least common denominator.
Show Step 2Now multiply each term by an appropriate quantity to get the least common denominator into the denominator of each term.
\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}} = \frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{{x\left( {3x + 8} \right)}}{{{{\left( {3x + 8} \right)}^2}\left( {3x + 8} \right)}}\] Show Step 3All we need to do now is do the addition and simplify the numerator of the result.
\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}} = \frac{{x + 10 + 3{x^2} + 8x}}{{{{\left( {3x + 8} \right)}^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3{x^2} + 9x + 10}}{{{{\left( {3x + 8} \right)}^3}}}}}\]