Question

√8-6i
Find the answer with steps​

Answer 1

Answer:

Begin with assuming that

(x+iy)2=8−6i  

with  x,y∈R  and equate the real and imaginary parts to obtain a system of two equations in two variables.

x2−y2+2ixy=8−6i  

⟹x2−y2=8  and  2xy=−6  

A nice way to solve this is to obtain  x2+y2 .

x2+y2=(x2−y2)2+(2xy)2−−−−−−−−−−−−−−−√  

⟹x2+y2=82+62−−−−−−√  

⟹x2+y2=10  

It is easy to solve for  x2  and  y2  now.

x2−y2=8,x2+y2=10  

⟹x2=9,y2=1  

To choose the correct sign while taking square roots, recall that  2xy=−6 , which means that  x  and  y  are of opposite signs.

⟹x=3,y=−1  or  x=−3,y=1

Hence, the square roots of  8−6i  are  ±(3−i) , i.e.  {3−i,−3+i} .

You can get one square root directly by using the following formula. (The other square root is obtained by negating this.)

z√=|z|+Rez2−−−−−−−−√+i⋅u(Imz)|z|−Rez2−−−−−−−−√,z∈C  

Here,  u  is the modified unit step function.

u(t)={−11t<0t≥0

Step-by-step explanation:

hope it is helpful