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Section 1.6 : Rational Expressions

12. Perform the indicated operation in the following expression.

\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}}\]

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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.

\[\frac{{x + 10}}{{{{\left( {3x + 8} \right)}^3}}} + \frac{x}{{{{\left( {3x + 8} \right)}^2}}} = \frac{{x + 10 + 3{x^2} + 8x}}{{{{\left( {3x + 8} \right)}^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3{x^2} + 9x + 10}}{{{{\left( {3x + 8} \right)}^3}}}}}\]