Problem 94 - Algebra and Trigonometry

Question

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\) is \(\frac{2+3x}{1+2x}\).

Step 1: Rearrange terms

Rearrange the terms in the numerator and denominator of the given expression to group together the terms with same exponent of \(x\). So, \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\) becomes \(\frac{\frac{2}{x^{2}}+\frac{3}{x}}{\frac{1}{x^{2}}+\frac{2}{x}}\).

Step 2: Combine like terms

Next, we can use the rule of adding fractions with same denominator. In our case \(x\) and \(x^{2}\) can be considered as our denominators. So, it becomes \(\frac{\frac{2+3x}{x^{2}}}{\frac{1+2x}{x^{2}}}\). As we have the same denominator for both fractions \(x^{2}\), fractions can be simplified to \(\frac{2+3x}{1+2x}\).

Step 3: Simplify + Alternative way

At this point, we have simplified the expression into the simplest form. An alternative way would be to find a common denominator for all fractions involved before starting simplification. So you would multiply \(\frac{3}{x}\) and \(\frac{2}{x^{2}}\) by \(x\) and \(x^{2}\) respectively similarly \(\frac{1}{x^{2}}\) and \(\frac{2}{x}\) by \(x^{2}\) and \(x\) respectively. Then add the fractions and simplify.