If (x+1/x) = 6, what is the value of x⁵ + 1/x⁵ = ?
Step-by-step explanation: x + 1/x = 5 ( given )
So,
x + 1/x = 5 …(1)
Squaring both sides
( x+1/x)² = 5²
x² + 1/x² + 2 =25
x² + 1/x² = 25 – 2
x² + 1/x ² = 23 ….(2)
Cube of the equation
x³ + 1/x³ = ( x+ 1/x )³ – 3 (x +1/x)
x³ + 1/x ³ = (5)³ – 3 (5)
x³ + 1/x³ = 125 -15
x³ + 1/x³ = 110 …(3)
Now taking all equation
(x² + 1/x² ) (x³ + 1/x³ ) = (x⁵ +1/x +x + 1/x⁵)
23 ₓ 110 = x⁵ + 1/x⁵
2530 = x⁵ + 1/x⁵