Math Problem Statement

Find the third roots of the complex number given by ((√2 + √6i)^10) / (4(2e^(iπ/3) - 2i)^7).

Solution

The image you provided contains a mathematical expression regarding the third roots of a complex number. To help you find the correct answer, I'll break down the steps for solving it:

The problem asks for the third roots of the complex number:

(2+6i)104(2eiπ/3−2i)7\frac{(\sqrt{2} + \sqrt{6}i)^{10}}{4(2e^{i\pi/3} - 2i)^{7}}

To find the third roots, we first need to express this complex number in polar form and then use De Moivre's theorem for taking the roots.

I'll compute the polar form of both terms in the expression and then find the cube roots. Let me process the calculations.

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Math Problem Analysis

Mathematical Concepts

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

Formulas

Polar form of a complex number: z = r e^(iθ)
De Moivre's Theorem: (r e^(iθ))^n = r^n e^(i nθ)
nth roots of a complex number: w_k = r^(1/n) e^(i(θ + 2kπ)/n), k = 0, 1, ..., n-1

Theorems

De Moivre's Theorem

Suitable Grade Level

Undergraduate Mathematics / Advanced High School

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