Question

X+1/x=3,x^5+1/x^5=? What is the answer

Answer 1

The answer is 123.

Explanation:

Given that-

(
`x + (1)/(x)` ) = 3.....................(i)

To find -

(
`x^(5) + (1 )/(x^(5) )` ) = ?

Now squaring both sides of equation (i)-

(
`x^(2)  + (1)/(x^(2) ) + 2` ) = 9

(
`x^(2) + (1)/(x^(2) )` ) = 7....................(ii)

On cubing equation (i) both sides-

(
`x^(3) + (1)/(x^(3) ) + 3 ( x + (1)/(x) )` ) = 27

(
`x^(3) + (1)/(x^(3) ) + 3 * 3` ) = 27

(
`x^(3) + (1)/(x^(3) )` ) = 27 - 9

(
`x^(3) + (1)/(x^(3) )` ) = 18.....................(iii)

Multiply equation (i) and equation (ii)-

(
`x^(2) + (1)/(x^(2) )` ) (
`x^(3) + (1)/(x^(3) )` ) = 7 × 18

(
`x^(5) + (1)/(x^(5) ) )+ ( x + (1)/(x) )` = 126

( [tex]x^{5} + \frac{1}{x^{5} } ) = 126 - 3

( [tex]x^{5} + \frac{1}{x^{5} } ) = 123