Question

If x=1/2-√3,find the value of (x+1/x)^3 -4

Answer 1

Answer:

60

Step-by-step explanation:

Given that ;

$\sf x = \dfrac{1}{2 - \sqrt{3} }$

At first, rationalise the denominator of x, by multiplying numerator and denominator by (2 + √3).

$\sf x = \frac{1}{2 - \sqrt{3} } \\ \\ \sf x = \frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3}} \\ \\ \sf x = \frac{2 + \sqrt{3} }{(2)^{2} - ( \sqrt{3}) {}^{2} } \\ \\ \sf x = \frac{2 + \sqrt{3} }{4 - 3} \\ \\ \sf x = 2 + \sqrt{3}$

Now, find the value of 1/x;

$\sf x = \frac{1}{2 - \sqrt{3} } \\ \\ \sf \implies \frac{1}{x} = 2 - \sqrt{3}$

Find the value of x + 1/x-

$\sf x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} \\ \\ \sf x + \frac{1}{x} = 2 + 2 \\ \\ \sf x + \frac{1}{x} = 4$

So,

$\sf (x + \frac{1}{x} )^{3} - 4 \\ \\ \implies \sf (4) {}^{3} - 4 \\ \\ \implies \sf 64 - 4 \\ \\ \implies \sf 60$

Hence, the answer is 60.