**If x = 2 – √3, then find the value of x**^{3} – 1/x^{3}

^{3}– 1/x

^{3}

**Answer :**

It is given that,

x = 2 – √3

so,

1/x = 1/(2 – √3)

By rationalizing the denominator, we get

= [1(2 + √3)] / [(2 – √3) (2 + √3)]

= [(2 + √3)] / [(2^{2}) – (√3)^{2}]

= [(2 + √3)] / [4 – 3]

= 2 + √3

Now,

x – 1/x = 2 – √3 – 2 – √3

= – 2√3

Let us cube on both sides, we get

(x – 1/x)^{3} = (-2√3)^{3}

x^{3} – 1/x^{3} – 3 (x) (1/x) (x – 1/x) = 24√3

x^{3} – 1/x^{3} – 3(-2√3) = -24√3

x^{3} – 1/x^{3} + 6√3 = -24√3

x^{3} – 1/x^{3} = -24√3 – 6√3

= -30√3

Hence,

x^{3} – 1/x^{3} = -30√3

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