Complex Numbers and Quadratic Equations

Express the following complex number in the form $a + ib$: $(5i)\left(-\frac{3}{5}i\right)$

Express the following complex number in the form $a + ib$: $i^9 + i^{19}$

Express the following complex number in the form $a + ib$: $i^{-39}$

Express the following complex number in the form $a + ib$: $3(7 + i7) + i(7 + i7)$

Express the following complex number in the form $a + ib$: $(1 - i) - (-1 + i6)$

Express the following complex number in the form $a + ib$: $\left(\frac{1}{5} + i\frac{2}{5}\right) - \left(4 + i\frac{5}{2}\right)$

Express the following complex number in the form $a + ib$: $\left[\left(\frac{1}{3} + i\frac{7}{3}\right) + \left(4 + i\frac{1}{3}\right)\right] - \left(-\frac{4}{3} + i\right)$

Express the following complex number in the form $a + ib$: $(1 - i)^4$

Express the following complex number in the form $a + ib$: $\left(\frac{1}{3} + 3i\right)^3$

Express the following complex number in the form $a + ib$: $\left(-2 - \frac{1}{3}i\right)^3$

Find the multiplicative inverse of the following complex number: $4 - 3i$

Find the multiplicative inverse of the following complex number: $\sqrt{5} + 3i$

Find the multiplicative inverse of the following complex number: $-i$

Express the following expression in the form of $a + ib$: $\frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})}$

Evaluate: $\left[i^{18} + \left(\frac{1}{i}\right)^{25}\right]^3$.

For any two complex numbers $z_1$ and $z_2$, prove that $\text{Re}(z_1 z_2) = \text{Re} z_1 \text{Re} z_2 - \text{Im} z_1 \text{Im} z_2$.

Reduce $\left( \frac{1}{1-4i} - \frac{2}{1+i} \right) \left( \frac{3-4i}{5+i} \right)$ to the standard form.

If $x - iy = \sqrt{\frac{a - ib}{c - id}}$ prove that $\left( x^2 + y^2 \right)^2 = \frac{a^2 + b^2}{c^2 + d^2}$.

If $z_1 = 2 - i, z_2 = 1 + i$, find $\left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + 1} \right|$.

If $a + ib = \frac{(x+i)^2}{2x^2 + 1}$, prove that $a^2 + b^2 = \frac{(x^2 + 1)^2}{2x^2 + 1}^2$.