Question Type 2: Finding the roots of complex numbers Exercises

Find all complex solutions of $z^2 = 5 + 12i$ and express them in the form $a + bi$.

Find all complex solutions of $z^4 = -8i$ and express them in polar form.

Find all complex solutions of $z^6 = -64$ and express them in the form $a+bi$.

Solve $z^6 = (4 + 4\sqrt{3}i)^3$ and express all six solutions in the form $a+bi$.

Find all complex solutions of $z^8 = -1$ and plot them on the Argand diagram.

Solve $z^7 = -128i$ and express all solutions in polar form.

Find all complex solutions of $z^6 = 64$ and express them in polar form.

Solve $z^4 = 16 + 16i$ and express all roots in polar form.

Find all complex solutions of $z^2 = 9$ and plot them on the Argand diagram.

Solve $z^5 = 32(\cos45^\circ + i\sin45^\circ)$ and express all roots in the form $a+bi$.

Find all complex solutions of $z^3 = 4 + 3i$ and plot them on the Argand diagram.

Find all complex solutions of $w^3 = (1 + i\sqrt{3})^5$ and express them in the form $a+bi$.

Determine all complex solutions of $z^4 = 16\bigl(\cos\tfrac{\pi}{3} + i\sin\tfrac{\pi}{3}\bigr)$ and plot them on the Argand diagram.