Question Type 6: Performing operations on variables of complex numbers Exercises

Determine the real value of $b$ such that the product $(2+bi)(4-3i)$ is a real number.

Simplify $(a+bi)+(4-3i)$.

Multiply $(2+ai)(3-2i)$ and express the result in the form $x+yi$.

Let $z=a+bi$ and $w=b+ai$. If $z^2+w^2=10i$, find all real $a$ and $b$.

Find all real values of $a$ such that $|a+2i|=5$.

Expand $(a+bi)^2$ and write the result in the form $x+yi$.

Determine the relationship between nonzero real numbers $a$ and $b$ such that $(a+bi)^3$ is a real number.

Find the conjugate of $a-bi$ and show that $(a-bi)(a+bi)$ is real.

Solve for real $a$ and $b$ if $(a+bi)^2=3-4i$.