Question

if x=3 - 2 √2 ,find x 3 - 1 / x3

Answer 1

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Answer 2

Answer:

$x^{3}-\frac{1}{x^{3}}$

=-140√2

Explanation:

Given x = 3-2√2 ---(1)

i) $\frac{1}{x}=\frac{1}{3-2\sqrt2}$

Rationalising the denominator we get,

= $\frac{3+2\sqrt{2}}{(3-2\sqrt{2})(3+2\sqrt{2})}$

= $\frac{3+2\sqrt{2}}{3^{2}-\left(2\sqrt{2}\right)^{2}}$

= $\frac{3+2\sqrt{2}}{9-8}$

= $3+2\sqrt{2}$---(2)

ii) x-1/x

= 3-2√2-(3+2√2)

= 3-2√2-3-2√2

= -4√2 ---(3)

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we know the algebraic identity:

a³-b³ = (a-b)³+3ab(a-b)

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Now ,

$x^{3}-\frac{1}{x^{3}}$

= $\left(x-\frac{1}{x}\right)^{3}+3\times x\times\frac{1}{x}\left(x-\frac{1}{x}\right)$

= (-4√2)³+3(-4√2)

= -128√2-12√2

= -140√2

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