Question

x4 +1/x4 =49 find x3 - 1/x3

Answer 1

x4+(1/x)4x4+(1/x)4needs to be first simplified. We know that

(x+(1/x))2=x2+(1/x)2+2−−−(i)(x+(1/x))2=x2+(1/x)2+2−−−(i)

and

(x2+(1/x)2)2=x4+(1/x)4+2−−−(ii)(x2+(1/x)2)2=x4+(1/x)4+2−−−(ii)

From above two equations, we can get x4+(1/x)4.x4+(1/x)4.

So now we need to find x+1/xx+1/x

For that get 1/x.Forwhichrationlizex.1/x.Forwhichrationlizex.

1/x=1∗(9−4root5)/(9+4root5)∗(9−4root5)1/x=1∗(9−4root5)/(9+4root5)∗(9−4root5)

1/x=9−4root51/x=9−4root5

x+1/x=9+4root5+9−4root5x+1/x=9+4root5+9−4root5

x+1/x=18x+1/x=18

Substituting its value in eq(i), we get

x^2 + (1/x)^2 = (18)^2 -2 = 322

substituting this value in eq(ii) , we get

x^4 + (1/x)^4 = (322)^2 - 2 = 103682

So, this is the final answer.