If x+(1/x)=5, then evaluate x6+(1/x6)
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Given : x + 1/x = 5
To find : x6 + 1/x6
( x3 + 1/x3 ) = (x + 1/x) (x2 + 1/x2 - 1) [Using the Identity : a3+b3 = (a+b)(a2+b2-ab) ]
= 5 (x2 + 1/x2 - 1) ------------------equation 1
To find x2+1/x2
(x+1/x)2 = x2 + 1/x2 + 2
52 = x2 + 1/x2 + 2
25 = x2 + 1/x2 + 2
Therefore x2 + 1/x2 = 25 - 2 = 23 ---------------equation 2
Substituting equation 2 in equation 1 :
x3+1/x3 = 5 (x2 + 1/x2 - 1)
= 5 (23-1) = 5 * 22 = 110
To find x6 + 1/x6:
(x3+1/x3)2 = x6 + 1/x6 +2 [Using Identity (a+b)2 = a2+2ab+b2]
1102 = x6 + 1/x6 +2
So, x6 + 1/x6 = 1102 - 2
= 12100-2
= 12098
Given : x + 1/x = 5
To find : x6 + 1/x6
( x3 + 1/x3 ) = (x + 1/x) (x2 + 1/x2 - 1) [Using the Identity : a3+b3 = (a+b)(a2+b2-ab) ]
= 5 (x2 + 1/x2 - 1) ------------------equation 1
To find x2+1/x2
(x+1/x)2 = x2 + 1/x2 + 2
52 = x2 + 1/x2 + 2
25 = x2 + 1/x2 + 2
Therefore x2 + 1/x2 = 25 - 2 = 23 ---------------equation 2
Substituting equation 2 in equation 1 :
x3+1/x3 = 5 (x2 + 1/x2 - 1)
= 5 (23-1) = 5 * 22 = 110
To find x6 + 1/x6:
(x3+1/x3)2 = x6 + 1/x6 +2 [Using Identity (a+b)2 = a2+2ab+b2]
1102 = x6 + 1/x6 +2
So, x6 + 1/x6 = 1102 - 2
= 12100-2
= 12098
