Find the four roots of the equation z^4 = + 8(sqrt(3) + i), in the form z = r*e^(i*theta). Draw the roots on an argand diagram.

This is a maths problem, which I believe is best taught by live demonstration and explanation while constantly promtpting the student to suggest further steps if they are able. If they are not, then you should gently prompt them towards further steps and carefully explain each line of working. If put into words, the basic outline of the solution to a problem like this is: 1) First find the modulus of z^4 by using pythagoras. Then find the argument of z^4 using triganometry, a simple argand diagram can often help to ensure that the sign of this value is correct. 2) Use the modulus and argument of z^4 to write it in exponential form. This makes taking the forth root of the equation to obtain z more simple. 3) Once z^4 is in exponential form, we need to add a arbitrary integer of 2pi to the argument in order to find all of the solutions. ie. Arg(z) = arg(z) +/- 2kpi. If the student is not convinced that this does not affect the value of the complex number, then it can be shown that they are the same by converting to polar form and asking them to substitutue some values in. 4) Take the 4th root of the equation. This is now simple because the right hand side is just an exponential. 5) Substitue in values of k, the argument of z should always fall between -pi and pi, so if a value of k gives an argument outside of this range then try a new value. Typically starting at k=0 and then trying k=+/- 1, then k = +/- 2 and so on.. is the best technique until you are more familiar with these types of problems. 6) In this case we want 4 roots sinse it is a forth order equation in z. 7) Now that the roots have been obtained, and are in exponential form. We can easily plot them on a argand diagram.