Questions 10 -13

Q10 Roots

Find the four roots of \((-3)^{1/4}\).

Strategy: This problem is the same as solving the equation \(w^4 = -3\) and as there are four roots they must form a square on an Argand diagram whose corners lie on a circle of radius \(3^{1/4}\). The roots of a negative number are sought so these must all be complex with a zero real part; i.e. with an imaginary part only.

Q11 Square roots

Find the square roots of \(i\), i.e. \(w^2 = i\). Find their magnitude and plot them on an Argand diagram.

Q12 Solve eqn.

Solve \(w^4 = 16\).

Strategy: Because the equation is fourth order, there are four solutions and not just the two real ones \(w = \pm 2\). Use the method of previous questions.

Q13 Modulus & argument

Calculate the modulus and argument of \(2 + 3i\) then calculate its square roots. What is the radius of the circle on which the roots lie and at what angles?

Strategy: The complex number \(2 + 3i\) is best converted into its trigonometric form to calculate the modulus and argument.