Multiplication of complex numbers can be obtained as multiplication of binomials. [math]\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd[/math] [br]If an [math]i=\sqrt{-1}[/math] is included on the second terms , [math]b[/math] and [math]d[/math] the result is[br] [math]\left(a+ib\right)\left(c+id\right)=ac+adi+bci+bdi^2=ac-bd+i\left(ad+bc\right)[/math] since [math]i^2=-1[/math].[br][br]A complex number can be represented as a magnitude and angle in the complex plane.[br][math]a+ib\Longrightarrow Rcos\theta+iRsin\theta\Longrightarrow R\angle\theta[/math][br]The relations [math]R^2=a^2+b^2[/math] and [math]tan\theta=\frac{b}{a}[/math] taking care of the quadrant.[br]Taking [math]c+id=Pcos\alpha+iPsin\alpha[/math] and calculating the product[br][math]\left(Rcos\theta+iRsin\theta\right)\left(Pcos\alpha+iPsin\alpha\right)=RP\left(cos\theta cos\alpha-sin\theta sin\alpha\right)+iRP\left(sin\theta cos\alpha+cos\theta sin\alpha\right)[/math][br][math]=RP\left(cos\left(\theta+\alpha\right)+isin\left(\theta+\alpha\right)\right)[/math][br][br]In other words to multiply in the complex plane multiply the magnitudes and add the angles.[br][br]In the following applet you can move the complex points 1 and 2 and the product is shown.

Raising a number to a power multiplies the angle by the power. Therefor an nth root will have n complex values. Each one multiplied n times would move completely around the circle. That is adding [math]\frac{2\pi}{n}[/math] to a root when raised to the power will result in an extra transversal of a full circle ending at the original spot.[br][br]The following applet shows the roots of a number between -2 and 2.[br]