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Algebraic Identities

Question

If $y - \frac{1}{y} = 9$ , find the value of $y^{3} - \frac{1}{y^{3}}$

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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 $x = y$ and $y = \frac{1}{y}, y^{3} - \frac{1}{y^{3}} = {(y - \frac{1}{y})}^{3} + 3 (y - \frac{1}{y})$
$= (9)^{3} + 3 (9) = 729 + 27 = 756$

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