Problem 94 - College Algebra Essentials

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 fraction is \( \frac{3x + 2}{1 + 2x} \) in both methods.

Step 1: Simplifying complex fraction by multiplying with the reciprocal

Complex fractions can be simplified by multiplying the numerator and the denominator by the same value to eliminate the smaller fractions. Here, \(x^{2}\) is the common denominator of all small fractions. Multiply the entire complex fraction by \(x^{2}\), Thus, \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\) becomes \(\frac{3x + 2}{1 + 2x}\)

Step 2: Check for further algebraic simplification

Examine the simplified fraction obtained in Step 1 for any further simplification possible. However, in this case \(\frac{3x + 2}{1 + 2x}\) cannot be simplified any further because neither numerator nor denominator has any common factors that can be cancelled out.

Step 3: Alternative Simplification: Treat the entire fraction as division operation

Another method to simplify complex fractions is to treat them as division. The fraction \(\frac{a}{b}\) can be seen as 'a divided by b'. So, \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\) can be treated as division of two fractions \((\frac{3}{x}+\frac{2}{x^{2}}) ÷ (\frac{1}{x^{2}}+\frac{2}{x})\).Using the rule that dividing by a fraction is the same as multiplying by its reciprocal, we end up with: \((\frac{3}{x}+\frac{2}{x^{2}}) * \frac{1}{(\frac{2}{x}+\frac{1}{x^{2}})}\). Multiply each term by \(x^{2}\) as it is the common denominator of all terms, which simplifies to: \(\frac{3x + 2}{1 + 2x}\)

Step 4: Check for further simplification with this method

The fraction obtained in this method is same as in method 1 and hence, no further simplification is possible.