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.