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Expansion of (a ± b ± c)²

Simplify: n-...

Question

Simplify: $(n - \frac{1}{n}) (n + \frac{1}{n}) (n^{2} + \frac{1}{n^{2}})$

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Solution

$(n - \frac{1}{n}) (n + \frac{1}{n}) (n^{2} + \frac{1}{n^{2}})$

$= (n (n + \frac{1}{n}) - \frac{1}{n} (n + \frac{1}{n}))$

$= (n^{2} + 1 - 1 - \frac{1}{n^{2}}) (n^{2} + \frac{1}{n^{2}})$

$= (n^{2} - \frac{1}{n^{2}}) (n^{2} + \frac{1}{n^{2}})$

$= n^{2} (n^{2} + \frac{1}{n^{2}}) - \frac{1}{n^{2}} (n^{2} + \frac{1}{n^{2}})$

$= n^{4} + 1 - 1 - \frac{1}{n^{4}}$

$= n^{4} - \frac{1}{n^{4}}$

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