Problem 54 - Introductory Algebra for College Students

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

In conclusion, the simplified form of \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\) is \((3x + 2) / (1 + 2x)\)

Step 1: Approach 1: Multiply by the LCD

Determine the least common denominator of the fractions, which is \(x^2\). Multiply both the numerator and denominator by \(x^2\), then distribute: \((x^2)\*(\frac{3}{x}+\frac{2}{x^{2}}) / (x^2)\*(\frac{1}{x^{2}}+\frac{2}{x})\) simplifies to \(3x + 2) / (1 + 2x)\)

Step 2: Approach 2: Separate the Fractions

Separate the fractions - both the numerator and denominator: \(\frac{\frac{3}{x}}{\frac{1}{x^{2}}+\frac{2}{x}} + \frac{\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\). Now simplify each one separately: first fraction will simplify to \(\frac{3x}{1} + \frac{2}{x}\), second fraction will simplify to \(\frac{2x}{1} + \frac{2}{x}\). Combining these results in \((3x + 2) / (1 + 2x)\).

Step 3: Verify the results

Verify that the results of both approaches are equivalent to each other.