Question

What are the solutions to x3=5−5i in polar form

a. 52‾√3cis(23π12)

b. 52‾√3cis(5π4)

c. 50‾‾‾√6cis(23π12)

d. 50‾‾‾√6cis(7π12)

e. 50‾‾‾√6cis(4π3)

f. 50‾‾‾√6cis(5π12)

g. 50‾‾‾√6cis(5π4)

h. 52‾√3cis(7π12)

i. 52‾√3cis(5π12)

Answer 1

The solutions to x^3 = 5 - 5i in polar form are √50cis(-π/12), √50cis(4π/3), and √50cis(5π/12).

Step-by-step explanation:

To find the solutions to the equation x^3 = 5 - 5i in polar form, we can start by converting the complex number 5 - 5i into polar form.

The magnitude (r) of the complex number can be found using the equation r = √(a^2 + b^2), where a is the real part and b is the imaginary part. In this case, the magnitude is √(5^2 + (-5)^2) = √(25 + 25) = √50.

To find the argument (θ), we can use the equation θ = tan^(-1)(b/a), where b is the imaginary part and a is the real part. In this case, the argument is tan^(-1)(-5/5) = tan^(-1)(-1) = -π/4.

Therefore, the complex number 5 - 5i in polar form is represented as √50cis(-π/4).

To find the solutions to x^3 = 5 - 5i, we can take the cube root of both sides of the equation. Using De Moivre's Theorem, we can express the cube root of the complex number in polar form as the cube root of the magnitude raised to the power of 3, and the argument divided by 3.

Therefore, the solutions to x^3 = 5 - 5i in polar form are:

a. √50cis(-π/12)

e. √50cis(4π/3)

i. √50cis(5π/12)