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Chapter 0: Problem 78

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

Short Answer

Expert verified
The simplified version of given expression using method 1 is \( \frac{5 + 3x^{2}}{3+2x^{2}} \) and using method 2 is \( 3 + \frac{2}{x} \) over \( 1+ \frac{2}{x^{2}} \).

Step by step solution

01

Method 1: Combine Like Terms

First, start by rearranging and combining like terms. Rewrite the expression as \( \frac{\frac{5}{x^{2}} + 3}{\frac{3}{x^{2}} + 2} \). From there, you could multiply every term by \( x^{2} \) to get rid of all denominators. This simplifies the equation to \( \frac{5 + 3x^{2}}{3+2x^{2}} \).
02

Method 2: Simplify Directly

The second method is to simplify directly. Start by multiplying both numerator and denominator by \( x^{2} \). The expression becomes \( \frac{3x + 2}{1+2x} \). From there, you can further simplify it to \( 3 + \frac{2}{x} \) over \( 1+ \frac{2}{x^{2}} \) if it is required to express the equation in terms of 1/x.

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