Math Problem Statement

Use the procedure in the previous example to find the complex values of z = (3√2 + 3i)^(1/3). Begin by obtaining the polar form (using the general form of the argument) of z^2. Write your answer in the form r(cos(θ + 2kπ) + i sin(θ + 2kπ)), e.g. do not evaluate cosine and sine. Now take this to the power 1/2 and use De Moivre's Theorem. Give your answers in decimal form, rounding to three decimal places where appropriate.

Solution

It looks like the image you've uploaded contains a question related to finding complex values using De Moivre's Theorem. The task appears to involve calculating the complex roots of a given number, specifically z=(3+2+3i)1/3z = (3 + \sqrt{2} + 3i)^{1/3}, by obtaining the polar form and using the general argument.

In order to solve this, we need to follow the steps outlined:

  1. Convert to Polar Form: The first part asks for expressing the complex number in polar form r(cosθ+isinθ)r (\cos \theta + i \sin \theta), where rr is the modulus and θ\theta is the argument (angle).

    • To find rr, calculate the modulus: r=x2+y2r = \sqrt{x^2 + y^2} where xx and yy are the real and imaginary parts, respectively.

    • To find the argument θ\theta, use the formula: θ=arg(z)=tan1(yx)\theta = \arg(z) = \tan^{-1}\left(\frac{y}{x}\right)

  2. Apply De Moivre's Theorem: De Moivre's Theorem states that for a complex number in polar form: [r(cosθ+isinθ)]n=rn(cos(nθ)+isin(nθ))\left[ r (\cos \theta + i \sin \theta) \right]^n = r^n (\cos(n\theta) + i \sin(n\theta)) For the cube roots, we set n=13n = \frac{1}{3} and calculate the three roots by varying k=0,1,2k = 0, 1, 2.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Form
De Moivre's Theorem
Roots of Complex Numbers

Formulas

r = √(x^2 + y^2)
θ = tan^(-1)(y/x)
De Moivre's Theorem: (r(cos(θ) + i sin(θ)))^n = r^n(cos(nθ) + i sin(nθ))

Theorems

De Moivre's Theorem

Suitable Grade Level

Grades 11-12