Given the equation x4−81=0 Because equation degree is equal to = 4 - contains the even number 4 in the numerator, then the equation has two real roots. Get the root 4-th degree of the equation sides: We get: 4x4=481 4x4=(−1)481 or x=3 x=−3 We get the answer: x = 3 We get the answer: x = -3 or x1=−3 x2=3
All other 2 root(s) is the complex numbers. do replacement: z=x then the equation will be the: z4=81 Any complex number can presented so: z=reip substitute to the equation r4e4ip=81 where r=3 - the magnitude of the complex number Substitute r: e4ip=1 Using Euler’s formula, we find roots for p isin(4p)+cos(4p)=1 so cos(4p)=1 and sin(4p)=0 then p=2πN where N=0,1,2,3,... Looping through the values of N and substituting p into the formula for z Consequently, the solution will be for z: z1=−3 z2=3 z3=−3i z4=3i do backward replacement z=x x=z