x + 1/x = 3 ........................ (i)
On squaring both side, we will get
(x + 1/x)2 = (3)2
Use the Square algebra formula (a + b)2 = a2 + 2ab + b2
⇒ x2 + 1/x2 + 2 = 9
⇒ x2 + 1/x2 = 7 ..................(ii)
Again squaring both sides, we will get
(x2 + 1/x2)2 = (7)2
Use the Square algebra formula (a + b)2 = a2 + 2ab + b2
⇒ x4 + 1/4 + 2 = 49
⇒ x4 + 1/x4 = 47 .....................(iii)
On cubing the equation (i) both side, we will get
(x + 1/x)3 = (3)3
Use the cube algebra formula (a + b)3 = a3 + 3a2b + 3ab2 + b3
⇒ x3+ 1/x3 + 3(x + 1/x) = 27
⇒ x3 + 1/x3 + 9 = 27 [∴ (x + 1/x) = 3]
⇒ x3 + 1/x3 = 18 ...................(iv)
On multiplying Eqs. (i) and (iii) , we get
∴ (x4 + 1/x4) (x + 1/x) = 47 x 3
⇒ x5 + 1/x5 + x3 + 1/x3 = 141
⇒ x5 + 1/x5 + 18 = 141 [from Eq. (iv)]
⇒ x5 + 1/x5 = 123
