Question

if x - 1/x=3+2√2,find x³-1/x³​

Answer 1

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Given :

• x-1/x = 3+2√2

To find :

• x³-1/x³

Solution :

$\sf{{x}^{3} - \frac{1}{ {x}^{3} } } = ( {x - \frac{1}{x}) }^{3} \\ \\ \sf{using \: {(a + b)}^{3} = {a}^{3} - {b}^{3} - 3ab(a + b) } \\ \\ \sf{{(x - \frac{1}{x}) }^{3} = {x}^{3} - {(\frac{1}{x}})^{3} - 3(x)( \frac{1}{x} ) } \\ \\ \sf{ = (x - {\frac{1}{x})}^{3} - 3( \cancel{x})( \frac{1}{ \cancel{x}} )} \\ \\ \sf{ = {(3 + 2\sqrt{2})}^{3} - 3} \\ \\ \sf{using \: {( a+ b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b) } \\ \\ \sf{ = [{3}^{3} + {(2 \sqrt{2})}^{3} + 3(3){(2 \sqrt{2})(3 + 2 \sqrt{2} }] - 3 } \\ \\ \sf{ = [ 27 + 16 \sqrt{2} + 18 \sqrt{2}(3 + 2 \sqrt{2}] - 3 }$