Question Type 3: Finding complex numbers with rational exponents Exercises

Find all values of $(4 + 3i)^{2/3}$ and write them in the form $a + bi$.

Express $1 + i$ in polar form and find all values of $(1 + i)^{1/2}$.

Write $2e^{i\frac{\pi}{6}}$ in $a + bi$ form and compute its $\frac{3}{2}$ power, giving all possible values.

Calculate all cube roots of $8i$ and express them in $a + bi$ form.

Find the two square roots of $-3 + 4i$ and express your answers in $a + bi$ form.

Find all values of $(2 - 2i)^{3/4}$ and plot the principal value on the Argand diagram indicating its argument.

Determine all cube roots of $3 + 3\sqrt{3}i$ and represent each on the Argand diagram with their arguments.

Compute $(-1 + i\sqrt{3})^{4/6}$, simplify the exponent and find all resulting values in $a + bi$ form.

Express $(5e^{i\pi/4})^{2/3}$ in $a + bi$ form for all values.

Express $(-8)^{2/3}$ in complex form and list all possible values.

Find all values of $(1 - \mathrm{i}\sqrt{3})^{2/3}$ in the form $a + b\mathrm{i}$, giving your answers to 3 significant figures.