Question

What are the fourth roots of −3+3 √3 i?

Answer 1

The fourth roots of -3 + 3√3i are 1 + i, -1 - i, -1 + i, and 1 - i.

Step-by-step explanation:

To find the fourth roots of -3 + 3√3i, we'll first express the complex number in polar form. Let z = -3 + 3√3i. The modulus of z, |z|, can be calculated as the square root of the sum of squares of the real and imaginary parts: |z| =
`√((-3)^2 + (3√3)^2)` = √(9 + 27) = √36 = 6. The argument of z, θ, can be found using the inverse tangent function: θ =
`tan^(-1)((3√3) / (-3)) = tan^(-1)(-√3)` = -π/3.

Now that we have z in polar form (z = 6 * (cos(-π/3) + i * sin(-π/3))), the fourth roots can be obtained by taking the nth root of the modulus and dividing the argument by n. For fourth roots, n = 4. So, the fourth root of z is given by
`6^{(1/4)` * [cos((-π/3 + 2kπ) / 4) + i * sin((-π/3 + 2kπ) / 4)], where k = 0, 1, 2, 3.

Substituting k = 0, 1, 2, 3 into the formula, we get:

1st root:
`6^{(1/4)` * [cos((-π/3) / 4) + i * sin((-π/3) / 4)] = 1 + i

2nd root:
`6^{(1/4)` * [cos((5π/12) + i * sin(5π/12)] = -1 - i

3rd root:
`6^{(1/4)` * [cos((7π/12) + i * sin(7π/12)] = -1 + i

4th root:
`6^{(1/4)` * [cos((11π/12) + i * sin(11π/12)] = 1 - i

Therefore, the fourth roots of -3 + 3√3i are 1 + i, -1 - i, -1 + i, and 1 - i.

Answer 2

To find the fourth roots of −3+3 √3 i, you must convert the complex number to polar form and then apply De Moivre's Theorem, dividing the argument by 4 and finding each root with an adjusted angle.

Step-by-step explanation:

The student has asked to find the fourth roots of the complex number −3+3 √3 i. We can perform this calculation by first expressing the complex number in polar form and then applying De Moivre's Theorem to find the roots.

To convert −3+3 √3 i to polar form, we must calculate the modulus and argument. The modulus is found using the formula r = √(a²+b²), which results in r = √(-3)²+(3√3)² = √(27). The argument (theta) is determined by taking the arctan of the imaginary part divided by the real part. Once we have the polar form, we divide the argument by the root we are seeking (fourth roots), and find each of the four roots using the formula r^(1/n) [cos(θ/4 + k*360°/n) + i sin(θ/4 + k*360°/n)], with k running from 0 to 3.

We must calculate these roots carefully, as the process involves multiple steps and the use of trigonometric functions. A clear understanding of complex numbers, polar form, and De Moivre's Theorem is essential for successfully finding the fourth roots of a complex number.