(-7+24I)^(1/4) . So that is the 4th root of (-7+24I) . The book I have does not explain in very good detail how to do this. I need to find all 4 roots.
How I tried to do it was by setting it equal to x. Then I made x^4=-7+24I . Next I try to pump this into the quadratic equation. To to this I make a=1, b=0, and c=7-24I . This gives me the roots of (-7+24I)^(1/2) but I am not sure how to use the quadratic to solve for powers higher than 2. I am thinking that a= 1 and -1 but I am not sure. When I do this I get 4 seperate answers, but they do not multiply out to -7+24I .
The easiest way of taking roots of complex numbers is to represent them in a polar form. If you’re unfamiliar with that notion, think of a 2D plane where the x-axis is a real number line, and the y-axis is an imaginary number line. All complex numbers can be represented as a point on this 2D plane: for example, (-7+24I) is the point (-7, 24), using cartesian coordinates.
Switch the cartesian representation (X + YI) into a polar form (Rcis[symbol]q[/symbol]) [“cis” = shorthand for Cos[symbol]q[/symbol] + I Sin[symbol]q[/symbol]] with the transformation R = SQRT(X[sup]2[/sup] + Y[sup]2[/sup]) and tan([symbol]q[/symbol]) = (Y/X). (-7+24I) = 25cis(73.74).
In this form, finding the nth root of Rcis[symbol]q[/symbol] is easy:
[R[sup]1/n[/sup] cis([symbol]q[/symbol]/n)][sup]n[/sup] = Rcis[symbol]q[/symbol],
so the fourth root of 25cis(73.74) is sqrt(5)cis(18.43). Remembering that 360 can be added to the angle [symbol]q[/symbol] without altering the fundamental location of the point gives the other three roots: sqrt(5)cis(108.43), sqrt(5)cis(198.43), and sqrt(5)cis(288.43), conveniently 90 degrees from each other. Converting these back to cartesian form and checking my work is an exercise left to the reader.
Alternatively, solve it in two steps. If z is a root of
z[sup]4[/sup] = -7+24i
write z[sup]2[/sup] = w = x + iy.
Then w[sup]2[/sup] = x[sup]2[/sup] - y[sup]2[/sup] + 2xyi, and so, equating real and imaginary parts
x[sup]2[/sup] - y[sup]2[/sup] = -7
xy = 12
Eliminating y or solving by inspection gives (x,y) = (3,4) or (-3,-4). So the square roots of -7+24i are 3+4i and -3-4i. To finish the question you now need to repeat this procedure to find the square roots of both of these ( you should find four solutions in all).
Ah HAH! THAT is what CIS means. I was looking online and saw that notation and wasnt sure what it meant. Both of your methods work and I did come up with the correct answer to the problem.
Any of you ever had to deal with Krysiegs Advanced Engineering Math book? Its a hard read, Ill tell you that much. Hard to teach yourself anything out of it.