Question

find fourth root of 1-i√3​

Answer 1

Step-by-step explanation:

Let x4=−1 so

x2=±i

If x2=i then, letting x=a+bi ,

(a+bi)2=i=a2−b2+2abi and so

a2=b2 , or a=±b and

2ab=1 .

If a=− b then −b2=1 which has no real solution for b .

If a=b then 2b2=1 so b=± 2–√/2 .

This means if x2=i then

x=±2–√2(1+i)

However, if x2=−i then, letting x=a+bi then by the same reasoning as before, a=±b and 2ab=−1 .

If a=b , then 2b2=−1 , which has no real solution.

If a=−b then −2b2=−1, and so b=±2–√2 .

This means that the 4 4th roots of -1 are:

x=2–√2−2–√2i , or

x=−2–√2+2–√2i

x=±2–√2(1+i)