Question

X³+1/X³ find the value if x=√3-√2

Answer 1

x=2+√3  
1/x = 1/ 2+√3
x+1/x =2+√3+1/(2+√3)
x+1/x  =[(2+√3)(2+√3)+1] /2+√3
  [by taking  LCM ]
x+1/x  =[(2+√3)² +1] /2+√3
x+1/x = (2² + √3² + 2×2 ×√3 )+1 / (2+√3)
[ (a+b)² = a² + b² + 2ab ]
x+1/x  = 4+ 3+ 4×√3 +1 /(2+√3)
x+1/x  = 7+1 + 4√3  
x+1/x = 8+4√3/ 2+√3
x+1/x =[8+4√3/(2 +√3)×[2-√3 / 2-√3]    
[by rationalising the denominator]
        =[8+4√3][2-√3] / 2²- √3
[ (a+b)(a - b) = a² - b² ]
        =16 + 8√3 - 8√3 - 4× 3 / 4 - 3
               =16 -12/1 = 4
x+1/x = 4…………… (1)

[x+1/x]³ = 4³    [On cubing both sides]
x³+1/x³+3x×1/x[x+1/x] = 64    
[using the formula  (x+y)³ = x³+ y³ + 3xy(x+y)]
x³ +1/x³+3[x+1/x] = 64    
x³ +1/x³+3×4 = 64     [from eq 1)
x³ +1/x³+ 12 = 64
x³ +1/x³ = 64 -12 = 52
x³ +1/x³ = 52

Hence, the value of x³ +1/x³ = 52

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