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Section 1.6 : Rational Expressions

9. Perform the indicated operation in the following expression.

\[\frac{2}{{3{x^2}}} - \frac{1}{{9{x^4}}} + \frac{2}{{x + 4}}\]

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Start Solution

We first need the least common denominator for this rational expression.

\[{\mbox{lcd : }}9{x^4}\left( {x + 4} \right)\] Show Step 2

Now multiply each term by an appropriate quantity to get the least common denominator into the denominator of each term.

\[\frac{2}{{3{x^2}}} - \frac{1}{{9{x^4}}} + \frac{2}{{x + 4}} = \frac{{2\left( {3{x^2}} \right)\left( {x + 4} \right)}}{{3{x^2}\left( {3{x^2}} \right)\left( {x + 4} \right)}} - \frac{{1\left( {x + 4} \right)}}{{9{x^4}\left( {x + 4} \right)}} + \frac{{2\left( {9{x^4}} \right)}}{{\left( {x + 4} \right)\left( {9{x^4}} \right)}}\] Show Step 3

All we need to do now is do the subtraction and addition then simplify the numerator of the result.

\[\frac{2}{{3{x^2}}} - \frac{1}{{9{x^4}}} + \frac{2}{{x + 4}} = \frac{{6{x^3} + 24{x^2} - \left( {x + 4} \right) + 18{x^4}}}{{9{x^4}\left( {x + 4} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{18{x^4} + 6{x^3} + 24{x^2} - x - 4}}{{9{x^4}\left( {x + 4} \right)}}}}\]