• If x = 2 + √3, find the value of x3+1/x3
Answers
Answer:
If x = 2 + √3, find the value of x3+1/x3
Step-by-step explanation:
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
HOPE THIS WILL HELP YOU...