Evaluate: `i^(135)` `(-sqrt(-1))^(4n+3),n in N` `sqrt(-25)+3sqrt(-4)+2sqrt(-9)`

Answer

8

Explanation

Given `(x^(3)+(a-10)x^(2)-x+a-6)/(x^(3)+(a-6)x^(2)-x+a-10)` reduces to a quotient of two linear functions.
Let `f(x)=x^(3)+(a-10)x^(2)-x+a-6=0 " has roots " alpha,beta,gamma`
Then `g(x)=x^(3)+(a-6)x^(2)-x+a-10=0 " has roots " alpha,beta,delta`
`therefore f(x)-g(x)=0 " has roots " alpha, beta`.
`therefore -4x^(2)+4=0rArrx=pm1`
`therefore alpha+beta=0, alphabeta=-1`
Now from `f(x)=0,alpha+beta+gamma=-(a-10)=10-a`
`rArr gamma =10-a" "(1)`
and `alpha beta gamma =6-a rArr gamma = a-6" "(1)`
From (1) and (2), `a=8`