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Chapter 0: Problem 94

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 form of the fraction \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\), using either method, is \(\frac{3x+2}{1+2x}\)

Step by step solution

01

Method 1: Simplification by finding common denominator

Find the common denominator for the fractions in both the numerator and the denominator. This common denominator is \(x^{2}\). So, the expression can be rewritten and simplified as \(\frac{3x+2}{1+2x}\)
02

Method 2: Simplification by clearing the fractions

To clear the fractions, the expression's numerator and denominator needs to be each multiplied by the greatest common integer multiple of \(x\) and \(x^{2}\), which is \(x^{2}\). This will result in the expression to be \(\frac{3x^{2}+2x}{x+2x^{2}}\). This can be rearranged to \(\frac{2x+3x^{2}}{2x^{2}+x}\). And this fraction just as the one obtained from method 1 can be reduced by dividing through by x, yielding \(\frac{3x+2}{1+2x}\)

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