Question

if x= 7+4√3 then find the value of x^3+1\x^3

Answer 1

x = 7 + 4√3

1/x = 1/(7 + 4√3)
=(7 - 4√3)/(7 + 4√3)(7 -4√3)
=(7 -4√3)/(7² -4√3²)
=(7 - 4√3)/(49 - 48)
=( 7 - 4√3)

hence, x + 1/x = (7 + 4√3) + (7 - 4√3) = 14

x³ + 1/x³ = (x + 1/x)³ -3(x + 1/x )
= ( 14)³ - 3(14)
= 14 × 193
= 2702

Answer 2

x = 7+4√3

1/x = 1/7+4√3

= 1/7+4√3 × 7-4√3/7-4√3

= 7-4√3/(7+4√3)(7-4√3)

= 7-4√3/(7²-(4√3)²)

= 7-4√3/(49-16(3))

= 7-4√3/(49-48)

1/x = 7-4√3
_______________
x + 1/x = 7+4√3 + 7-4√3

(x + 1/x) = 14
_________________
x³ + 1/x³ = (x+1/x)³ - 3(x)(1/x) {x+1/x)

= (14)³ - 3(14)

= 2744 - 42

= 2702

Hope it helps