In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. _x³ − (1 + i√3 = 0

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1. Rewrite the equation as \( x^3 = 1 + i\sqrt{3} \).
2. Express the complex number \( 1 + i\sqrt{3} \) in polar form. Calculate the modulus \( r = \sqrt{1^2 + (\sqrt{3})^2} \) and the argument \( \theta = \tan^{-1}(\frac{\sqrt{3}}{1}) \).
3. Use De Moivre's Theorem to find the cube roots of the complex number. The formula is \( x = r^{1/3} \left( \cos\left(\frac{\theta + 2k\pi}{3}\right) + i\sin\left(\frac{\theta + 2k\pi}{3}\right) \right) \) for \( k = 0, 1, 2 \).
4. Calculate the three cube roots by substituting \( k = 0, 1, 2 \) into the formula from the previous step.
5. Convert each of the roots from polar form back to rectangular form using \( x = r(\cos\theta + i\sin\theta) \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the imaginary part. They can be represented graphically on the complex plane, with the x-axis representing the real part and the y-axis representing the imaginary part. Understanding complex numbers is essential for solving equations that involve them, as they extend the number system beyond real numbers.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ) relative to the positive real axis, represented as r(cos θ + i sin θ) or re^(iθ). This form is particularly useful for multiplication and division of complex numbers, as well as for finding roots, as it simplifies the calculations involved. Converting between rectangular and polar forms is a key skill in complex number analysis.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power of the complex number can be expressed as r^n(cos(nθ) + i sin(nθ)). This theorem is also used to find the nth roots of complex numbers, which is crucial for solving polynomial equations in the complex number system. Understanding this theorem allows for efficient computation of powers and roots of complex numbers.