Surds & Rectangular Form — practice questions

Simplify $\sqrt{98} + \sqrt{32}$, giving your answer in the form $a\sqrt{b}$ where $b$ is as small as possible.

Express $\dfrac{5}{2\sqrt{3} - 1}$ with a rational denominator, giving your answer in simplest form.

Solve $\sqrt{x + 7} - \sqrt{x} = 1$ for $x$, showing that your solution is valid.

Given $z = 4 - 3i$ and $w = -1 + 5i$, find $z + w$ and $zw$, giving each answer in the form $a + bi$.

Find the real number $k$ for which $(2 + ki)^2$ is purely imaginary.

The complex number $z = x + yi$, where $x$ and $y$ are real, satisfies $z^2 = \overline{z} \cdot 4i$ where $\overline{z} = x - yi$. Find all possible values of $z$.

Express $\dfrac{2 + 6i}{1 + i}$ in the form $a + bi$.

Find the real number $k$ such that $\dfrac{k + 3i}{2 - i}$ is purely real.

The complex number $z$ satisfies $\dfrac{z}{z + 2} = 1 - i$. Find $z$, and hence find the real number $m$ for which $mz + \overline{z}$ is purely imaginary.