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Algebraic Identities

Question

If $x + \frac{1}{x} = 5$ , find the value of $x^{3} + \frac{1}{x^{3}}$

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Solution

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We know that $x^{3} + y^{3} = (x + y)^{3} - 3 x y (x + y)$
Put $y = \frac{1}{x}, x^{3} + \frac{1}{x^{3}} = {(x + \frac{1}{x})}^{3} - 3 (x + \frac{1}{x})$
$= (5)^{3} - 3 (5) = 125 - 15 = 110$

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