Problem 106 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of the operation \[ \frac{1}{x^{n} - 1} - \frac{1}{x^{n} + 1} - \frac{1}{x^{2n} - 1} \] is \[ \frac{2x^{2n}}{(x^{n} - 1) (x^{n} + 1) (x^{2n} - 1)} \]

Step 1: Identify Common Denominator

For the operation to successfully be performed all the fractions need to have a common denominator. Looking at the denominators \(x^{n} - 1\), \(x^{n} + 1\), and \(x^{2n} - 1\), the common denominator will be their product, that is \( (x^{n} - 1) (x^{n} + 1) (x^{2n} - 1) \).

Step 2: Adjust each fraction

Now, each fraction needs to be adjusted by multiplying the numerator and denominator of each fraction by the missing terms in the common denominator. That is: \[ \frac{ (x^{n} + 1)(x^{2n} - 1)}{ (x^{n} - 1) (x^{n} + 1) (x^{2n} - 1)} - \frac{ (x^{n} - 1)(x^{2n} - 1)}{ (x^{n} - 1) (x^{n} + 1) (x^{2n} - 1)} - \frac{(x^{n} - 1) (x^{n} + 1)}{(x^{n} - 1) (x^{n} + 1) (x^{2n} - 1)} \]

Step 3: Simplify the numerator

Subtract the numerators of the fractions, as they now have the same denominator. This now equates to: \[ \frac{(x^{n} + 1)(x^{2n} - 1) - (x^{n} - 1)(x^{2n} - 1) - (x^{n} - 1)(x^{n} + 1)}{(x^{n} - 1) (x^{n} + 1) (x^{2n} - 1)} \].

Step 4: Further simplify

On simplifying the numerator further, the completed operation expression will be: \[ \frac{2x^{2n}}{(x^{n} - 1)(x^{n} + 1)(x^{2n} - 1)} \]