If `(2+sqrt(3))^n=I+f,` whare `I` and `n` are positive integers and `0

If (5+2sqrt(6))^(n)=m+f, where n and m are positive integers and 0<=f<=1, then (1)/(1-f)-f is equal to

If (9 + 4 sqrt(5))^(n) = I + f ,n and l being positive integers and f is a proper fraction , show that (I-1 ) f + f^(2) is an even integer.

" If "(4+sqrt(15))^(n)=I+F" when "I," n are positive integers,"0ltFlt1" then "(I+F)(1-F)

(-i)^(4n+3), where n Is a positive integer.

Show that (-sqrt(-1))^(4n+3) =i , where n is a positive integer.

If (9 + sqrt80)^n =1+f, where I, n are integers and 0 lt f lt 0 : (A) I is an odd Integer (B) I is even integer (C) (I+f) (1-f)=1 (D) 1-f=(9-sqrt 80)^n

If (sqrt(3)+i)^(n)=x_(n)+iy_(n) and n is a positive integer then the value of x_(n-1)y_(n)-x_(n)y_(n-1)=

((1+i)/(1-i))^(4n+1) where n is a positive integer.

Prove that (a)(1+i)^(n)+(1-i)^(n)=2^((n+2)/(2))*cos((n pi)/(4)) where n is a positive integer. (b) (1+i sqrt(3))^(n)+(1-i sqrt(3)^(n)=2^(n+1)cos((n pi)/(3)), where n is a positive integer