Question

Which expression is a fourth root of -1 + i√3?
A. (2)^(1/4)(cos(320º) + isin(320º))
B. (2)^(1/4)(cos(30º) + isin(30º))
C. (2)^(1/4)(cos(60º) + isin(60º))
D. (2)^(1/4)(cos(280º) + isin(280º))

Answer 1

You could put your expression into exponential form, or just raise each answer to the 4th power and see which one is equivalent to, $$-1+i\sqrt{3}$$

Answer 2

My answer is B.

Answer 3

the use of polar conversions is the fastest way to calculate nth roots of ANY number

Answer 4

Please show the steps you took to reach B (for the edification of others).

Answer 5

let z = -1 + i√3
|z|^2 = 1 + 3 = 4 => |z| = 2

z = |z| exp(iy) = 2 exp(iy) = 2(cos(y) + i sin(y))

where cos(y) = - 1/2, sin(y) = √3/2

sin(60) = √3/2 but since cos(y) is negative y = 180 - 60 = 120 degrees

so: z = 2 exp(i 120º)

z^1/4 = +/- 2^1/4 exp(i 120º/4) = +/- 2^1/4 exp(i 30º)

Answer 6

1. convert to polar form by calculating the magnitude and angle. to do so, take the square root of (a^2 + b^2) in the original expression, and you have the magnitude. then, to find the angle, simply take the inverse tangent of b/a, in the original expression.

now you have the polar form r*cis(theta), where r is the magnitude, and theta is the angle. if you aren't familiar with the cis notation, it is simply the compressed form of
(cos(theta) + i*sin(theta))

2. now that you have your polar form of the expression, set it equal to some number z, such that z = r*cis(theta).

what you want is the fourth root of z, so simply take the fourth root of both sides of the equation.

3. by DeMoivre's theorem, to find all the fourth roots of the expression, take the fourth root of r, and multiply it with cis((theta + 360k)/4), where k = 0, 1, 2, or 3.

now all that is left is to substitute k for 0, 1, 2, or 3, to find all the fourth roots, and expand the entire expression. whichever one matches an answer choice is the correct answer.

Answer 7

\[2^{(1/4)\cos 30^{o})}+i \sin (30^{o}))\]
Do I have the answer B. copied correctly, there seems to be one extra patenthese at the end. And thank you both while I attempt to duplicate your steps.

Answer 8

B. \[\sqrt[4]{2}(\cos(30º) + isin(30º))\]

Answer 9

Thanks @ilfy214 I have it now.

Answer 10

No problem! :)