IB Mathematics Analysis and Approaches 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 $a+bi$.