IB Mathematics Analysis and Approaches 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.