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Chapter 7: Problem 54

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

Short Answer

Expert verified
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 by step solution

01

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 \(3x + 2) / (1 + 2x)\)
02

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 \((3x + 2) / (1 + 2x)\).
03

Verify the results

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

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Most popular questions from this chapter

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