Consider the equation [math]x=1[/math]. Clearly there is only [color=#666666][size=200]one [/size][/color]solution here![br][br][br]Now consider [math]x^2=1[/math] which we can express as [math]\sqrt{1}[/math]. We know there are [color=#666666][size=200]two [/size][/color]solutions: [math]x=1[/math] and [math]x=-1[/math][br][br][br]Now consider [math]x^3=1[/math] which again we can express as [math]\sqrt[3]{1}[/math]. From The Fundamental Theorem of Algebra we know that there must be [color=#666666][size=200]three [/size][/color]solutions but we can only think of 1? Surprise, surprise the other two solutions lie in the complex plane.[br][br][br]Experiment with the activity below. You should notice some nice patterns that emerge:[br][list][*]Firstly, roots occur at equal rotations to each other - think spokes on a bicycle where the number of [size=200][color=#999999]"spokes" [/color][/size]is equal to the [size=200][color=#999999]number of roots[/color][/size][/*][*]All roots occur within a [color=#999999][size=200]full rotation[/size][/color] of a circle so the spacing between each root should be [math]360^\circ\div n[/math] or [math]2\pi\div n[/math][br][/*][*]For values other than 1 (e.g. [math]x^n=2[/math] or [math]x^n=-5[/math]) the radius of the circles [size=200][color=#999999]decrease [/color][/size](or increase if between -1 and 1) by a factor of the [math]n[/math]th root (look at the green circles - each time you raise [i]n[/i] you are effectively taking another root).* [/*][/list][size=85][i]*This makes sense... Think about [/i][math]x^1=2[/math][i] clearly there will just be one root that lies on the circle with radius 2.[br][br]Now think about [/i][math]x^2=2[/math][i] which will give two roots but each one will lie on the circle with radius [/i][math]\approx1.414[/math][i] since [/i][math]\sqrt{2}=1.414...[/math][i][br][br]Now think about [/i][math]x^3=2[/math][i] which will give three roots but each one will lie on the circle with radius [/i][math]\approx1.260[/math][i] since [/i][math]\sqrt[3]{2}=1.260...[/math][i][br][br]And the same thing for [/i][math]\sqrt[4]{2}[/math][i] and [/i][math]\sqrt[5]{2}[/math][i] etc.[br][br]So the size of the radius is determined by the amount of roots you've taken.[/i][/size]

Notice that the same is true for complex numbers...