The roots of the equation z^4+1=0 are (A) (±1 ±i) (B) (±2 ±2 i) (C) (1)/(√(3))(±1 ±i) (D) None o

step 1 given question we are given an equation which states that z to the power 4 plus 1 equal to 0 fro

The roots of the equation z^4+1=0 are (A) (±1 ±i) (B) (±2 ±2...

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Key Concepts

Nth Roots of Complex Numbers

Finding the nth roots of a complex number involves expressing the number in polar form and then applying the formula z = r^(1/n)·e^(i(?+2?k)/n) for k = 0, 1, ..., n-1. This method reveals that the roots are evenly distributed in angle around a circle in the complex plane, leading to symmetric patterns and insights into the structure of polynomial equations.

De Moivre's Theorem

De Moivre's Theorem provides a formula linking complex numbers and trigonometry, stating that (cos? + i sin?)^n = cos(n?) + i sin(n?) for any integer n. This theorem is particularly useful for finding powers and roots of complex numbers when they are expressed in polar form.

Complex Numbers

Complex numbers expand the real number system by including the imaginary unit i (where i² = -1), allowing solutions to equations that have no real roots. They are typically expressed in the form a + bi and play a fundamental role in many areas of mathematics, particularly in analyzing polynomial equations and oscillatory phenomena.

Polar Representation

The polar form of a complex number expresses it in terms of its magnitude (modulus) and its angle (argument) from the positive real axis. This form, usually written as r·(cos? + i sin?) or r·e^(i?), simplifies the multiplication, division, and extraction of roots of complex numbers.

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