Question

Find the three cube roots of the following complex number:

(3−4i)
a. (−0.364+1.67i),(−1.26−1.15i)&(1.63−0.52i)
b. (−0.364+1.67i),(−1.26−1.15i)&(2.61−0.45i)
c. (−0.364+1.67i),(−1.26+1.15i)&(1.63−0.52i)
d. (−0.364−1.67i),(−1.26−1.15i)&(1.63−0.52i)
​

Answer 1

To find the cube roots of a complex number, we convert it to polar form, apply De Moivre's Theorem, and then convert the solutions back to rectangular form.

Step-by-step explanation:

The student is asking to find the three cube roots of the complex number (3−4i). To find the cube roots of a complex number, we can convert the complex number into polar form and then apply De Moivre's Theorem. The polar form of a complex number is r(cosθ + isinθ), where r is the magnitude and θ (theta) is the argument of the complex number. Once we have the polar form, we can find the nth roots by taking the nth root of the magnitude and dividing the argument by n, and then apply the formula for each of the n roots by multiplying the argument by 2πk/n, where k = 0, 1, ..., n-1.

In this case, we first need to find the magnitude ℓ which is the square root of the sum of the squares of the real and imaginary parts and the argument θ which can be found using the arctan function. With the cube roots, n=3, and we will use k=0, 1 and 2 to find the three cube roots of (3−4i). After finding the cube roots in polar form, we'll convert them back to rectangular form to get the final answers.

Answer 2

The cube roots of a complex number can be found using the formula (a + bi)^(1/3) = r(cos((theta + 2*pi*n)/3) + i*sin((theta + 2*pi*n)/3)). For the complex number (3-4i), the cube roots are (-0.364 + 1.67i), (-1.26 - 1.15i), and (1.63 - 0.52i).

Step-by-step explanation:

The cube roots of a complex number can be found by using the formula:

(a + bi)^(1/3) = (r(cos((theta + 2*pi*n)/3) + i*sin((theta + 2*pi*n)/3)))

where a+bi is the complex number, r = |a+bi| is the modulus of the complex number, and theta = arg(a+bi) is the argument of the complex number.

Let's find the cube roots of the complex number (3-4i):

Step 1: Calculate the modulus of the complex number:

|3-4i| = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5

Step 2: Calculate the argument of the complex number:

arg(3-4i) = arctan((-4)/3) = -53.13 degrees or -0.93 radians

Step 3: Use the formula to find the three cube roots:

(3-4i)^(1/3) = 5^(1/3) * [cos((-0.93 + 2*pi*n)/3) + i*sin((-0.93 + 2*pi*n)/3)]

where n = 0, 1, 2

Substituting the values of n, we get the three cube roots of the complex number:

(-0.364 + 1.67i), (-1.26 - 1.15i), (1.63 - 0.52i)