Show that (-(1)/(2)+(√(3) i)/(2))^3=1.

Name: Problem 30, Chapter 31: Algebra & Trigonometry Problem Solver
Uploaded: 2020-07-29T19:32:12-08:00
Duration: 1 min 39 s
Channel: Rakesh Kumar Sharma
Description: Show that (-(1)/(2)+(√(3) i)/(2))^3=1.

Key Concepts

Square Roots in the Complex Plane

Finding the square root of a complex number involves expressing the number in polar form and then taking the square root of its modulus while halving its argument. This process is important for correctly determining all possible square roots, which may lead to multiple valid solutions in the context of complex arithmetic.

De Moivre's Theorem

De Moivre's theorem states that for any complex number expressed in polar form and any integer power, raising the number to that power involves raising the modulus to that power and multiplying the argument by the power. This theorem is essential for simplifying computations involving the raising of complex numbers to powers.

Roots of Unity

Roots of unity are the solutions to the equation z^n = 1 in the complex plane. They are geometrically represented as points evenly spaced around the unit circle. Understanding roots of unity is critical in problems involving powers and symmetries of complex numbers, particularly when showing that a complex expression raised to a certain power equals one.

Complex Numbers

Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane by introducing an imaginary unit. They can be expressed in rectangular form (a + bi) or in polar form, which is particularly useful for multiplication, division, and exponentiation.

Polar Representation of Complex Numbers

The polar form of a complex number expresses it as a product of its modulus (or absolute value) and an exponential involving its argument (or angle). This form simplifies operations such as powering and rooting by converting multiplicative and exponentiation operations into addition and multiplication of angles, respectively.

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