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 Solution
We 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 2
Now 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 3
All we need to do now is do the addition and simplify the numerator of the result.
\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}} = \frac{{x + 10 + 3{x^2} + 8x}}{{{{\left( {3x + 8} \right)}^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3{x^2} + 9x + 10}}{{{{\left( {3x + 8} \right)}^3}}}}}\]