高中數學. 請問大大, 請問下題如何解呢?謝謝 題目 : i 的平方根?

求x^2=i (可知道x一定為虛數[只有虛數*虛數會產生虛數])

故令x=a+bi，(a+bi)^2=i
a^2+2abi-b^2=i (實部=實部，虛部=虛部)
可得
a^2-b^2=0 => a^2=b^2 => a=b...(1)
2ab=1...(2)
(1)帶入(2)
2a^2=1
=>a^2=1/2
=>a=±1/√2=±√2/2
b=a=±√2/2

故x=a+bi=±√2/2±√2/2i

Sol
x^2=i=0+i=Cos(π/2)+iSin(π/2)=Cos[(2n+1)π/2]+iSin[(2n+1)π/2]
x=Cos[(2n+1)π/4]+iSin[(2n+1)π/4]
(1) n=0
x=Cos(π/4)+iSin(π/4)=√2/2+i√2/2
(2) n=1
x=Cos(3π/4)+iSin(3π/4)=－√2/2－i√2/2