If x=2+√3 , find the value of x³ + 1/x³.
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x=2+3 [given]
x+1/x =2+∫3+1/2+∫3
=[2+∫3][2+∫3]+1/2+∫3 [taking LCM ]
=[2+∫3]2+1/2+∫3
=4+3+[4×∫3]+1/2+∫3
=8+4∫3/2+∫3
=[8+4∫3/2+∫3]×[2-∫3/2-∫3] [by rationalising the denominator]
=[8+4∫3][2-∫3]/[2]2-[∫3]2
=16-8∫3+8∫3-4[∫3]2
=16-12
x+1/x=4
[x+1/x]3=43 [cubing both side]
x3+1/x3+3×x×1/x[x+1/x]=64 [using {x+y}3]
x3+1/x3+3×4=64 [substituting the value of x+1/x]\
x3+1/x3=64-12
x3+1/x3=52 [Ans]
I HOPE ITS HELP YOU
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