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Chapter 0: Problem 117

Perform the indicated operations. $$\frac{1}{x^{n}-1}-\frac{1}{x^{n}+1}-\frac{1}{x^{2 n}-1}$$

Short Answer

Expert verified
The simplified form of the given expression is \( \frac{1}{x^{2n}-1} \)

Step by step solution

01

Combine the fractions

The key to combining fractions is finding a common denominator. For the given fractions, the common denominator is \(x^n (x^n-1)(x^n+1)(x^{2n}-1)\). Use the formula: \(\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\) to rewrite the fractions with this common denominator.
02

Combine \(1/(x^{n}-1)\) and \(1/(x^{n}+1)\)

We can rewrite \(\frac{1}{x^n-1} - \frac{1}{x^n+1}\) as \(\frac{x^n+1 - (x^n -1)}{(x^n -1)(x^n +1)}\) which simplifies further to \(\frac{2}{x^{2n}-1}\).
03

Subtract \(1/(x^{2n}-1)\) from the result obtained in Step 2

The resulting fraction is \(\frac{2}{x^{2n}-1} - \frac{1}{x^{2n}-1} = \frac{2-1}{x^{2n}-1}\), which simplifies further to \(\frac{1}{x^{2n}-1}\).

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Most popular questions from this chapter

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

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