Question Type 5: Applying multiple operations onto a complex number Exercises

Given $z = (1 + 2i)^3$, expand and express $z$ in the form $a + bi$.

Given $z = \frac{(1 + i)^2}{2 - i} + 3i$, simplify $z$ and express it in the form $a + bi$.

Find the complex conjugate of $z = \frac{2 - i}{3 + 4i}$ and simplify the result to the form $a + bi$.

Divide $5 + 2i$ by $1 - 2i$, then subtract $i$, and express the result in the form $a + bi$.

Simplify $\left(\frac{2 + 3i}{-1 + 4i}\right) \cdot (1 - 2i)$ and express your answer in the form $a + bi$.

Find the modulus and argument (in radians) of $z = \frac{2 + 3i}{1 - i}$.

On an Argand diagram, represent the complex number $z = \left(\frac{1 - i}{2 + i}\right)^2$ and find its modulus and argument.

Simplify $\frac{(2 + i) - (5 + 3i)}{3 + 2i}$ and express your answer in the form $a + bi$.

Compute $(3 - 4i)(1 + i) + (2 + i)^2$ and express your answer in the form $a + bi$.

Simplify $\frac{(2 + i)^3 - (1 - i)^3}{1 + i}$ and express your answer in the form $a + bi$.