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Chapter 0: Problem 68

Simplify each complex rational expression. $$\frac{\frac{x}{x-2}+1}{\frac{3}{x^{2}-4}+1}$$

Short Answer

Expert verified
The simplified form of the given complex rational expression is \(\frac{2x-2}{3 + x^3 - 4x}\).

Step by step solution

01

Simplify the Numerator

First, the aim is to simplify the numerator by acquiring a common denominator for the fractions involved. The common denominator here is \(x(x-2)\). So, the numerator becomes \(\frac{x + (x-2)}{x(x-2)} = \frac{2x-2}{x(x-2)}\).
02

Simplify the Denominator

Moving on to simplify the denominator. The common denominator for the fractions involved in the denominator is \(x(x^{2}-4)\), which can be factored as \(x(x+2)(x-2)\). Consequently, the denominator becomes \(\frac{3 + x(x+2)(x-2)}{x(x+2)(x-2)} = \frac{3 + x^3 - 4x}{x(x+2)(x-2)}\). Therefore, the overall complex fraction becomes \(\frac{2x-2}{x(x-2)} \div \frac{3 + x^3 - 4x}{x(x+2)(x-2)}\).
03

Dividing Fractions

The operation of dividing fractions can be simplified by multiplying the first fraction by the reciprocal of the second. This means, flipping the second fraction and changing the operation from division to multiplication. So, the operation becomes \(\frac{2x-2}{x(x-2)} \times \frac{x(x+2)(x-2)}{3 + x^3 - 4x}\).
04

Simplify Further

Simplify further by cancelling out the common terms in the numerator and the denominator. The common terms are \(x(x-2)\) which results in \(\frac{2x-2}{3 + x^3 - 4x}\). Thus, this is the simplified form of the given complex rational expression.

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Most popular questions from this chapter

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$

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