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If x = 2 + √3, find the value of x3 + \(\frac{1}{x^3}\).

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Given x = 2 + √3 and given to find the value of x3\(\frac{1}{x^3}\)

We have x = 2 + √3

rationalising factor for denominator is 2 - √3

and also \((x + \frac{1}{x})\) = 2 + √3 + 2 - √3

= 2 + 2 = 4

∴ \((x + \frac{1}{x})\) = 4 equation (i)

We know that,

By putting \((x + \frac{1}{x})\) = 4 we get

∴ The value of  x3\(\frac{1}{x^3}\) is 52.

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