Section 1
Complex numbers, sums, products and quotients
Express each of the following numbers in the form $a+b i$.$$5+\sqrt{-4}$$
Express each of the following numbers in the form $a+b i$.$$7-\sqrt{-7}$$
Express each of the following numbers in the form $a+b i$.$$-6$$
Express each of the following numbers in the form $a+b i$.$$-\sqrt{49}$$
Express each of the following numbers in the form $a+b i$.$$\sqrt{-81}$$
Express each of the following numbers in the form $a+b i$.$$-\sqrt{\frac{-25}{16}}$$
Perform the following operations and express your answer in the form $a+b i$.$$(-3+4 i)+(2-5 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$(-3+4 i)-(2-5 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$(-3+4 i)(2-5 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$3 i-(2-4 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$(2-7 i)(3+4 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$(1+i)(2-3 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{3+2 i}{2+5 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{2-i}{3+2 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$\left(\frac{2}{3}-\frac{1}{2} i\right)+\left(\frac{1}{3}+\frac{1}{2} i\right)$$
Perform the following operations and express your answer in the form $a+b i$.$$\left(\frac{2}{3}-\frac{1}{2} i\right)\left(\frac{2}{3}+\frac{1}{2} i\right)$$
Perform the following operations and express your answer in the form $a+b i$.$$\left(\frac{2}{3}-\frac{1}{2} i\right) \div\left(\frac{1}{3}+\frac{1}{2} i\right)$$
Perform the following operations and express your answer in the form $a+b i$.$$(2+i)(3-2 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{1}{i}(3-7 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$(2+5 i)-(-2-5 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{13}{5-12 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{12 i}{3+4 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$3 i\left(3-\frac{2}{3} i\right)$$
Perform the following operations and express your answer in the form $a+b i$.$$(3+5 i)(6-10 i)$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{39-52 i}{24+10 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$(7-4 i)^{-1}$$
Perform the following operations and express your answer in the form $a+b i$.$$(5-12 i)^{-1}$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{3}{3-4 i}+\frac{2}{6+8 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{(7+8 i)(2-5 i)}{5-12 i}$$
Perform the following operations and express your answer in the form $a+b i$.$$\frac{5-\sqrt{-144}}{3+\sqrt{-16}}$$
$(2+y i)(x+i)=1+3 i,$ where $x$ and $y$ are real numbers. Solve for $x$ and $y$
a) Evaluate $(1+i \sqrt{3})^{3}$b) Prove that $(1+i \sqrt{3})^{6 n}=8^{2 n},$ where $n \in Z^{+}$c) Hence, find $(1+i \sqrt{3})^{48}$
a) Evaluate $(-\sqrt{2}+i \sqrt{2})^{2}$b) Prove that $(-\sqrt{2}+i \sqrt{2})^{4 k}=(-16)^{k},$ where $k \in Z^{+}$c) Hence, find $(-\sqrt{2}+i \sqrt{2})^{46}$
If $z$ is a complex number such that $|z+4 i|=2|z+i|,$ find the value of $|z|$ $(|z|=\sqrt{x^{2}+y^{2}} \text { where } z=x+i y .)$
Find the complex number $z$ and write it in the form $a+$ bi if $z=3+\frac{2 i}{2-i \sqrt{2}}$
Find the values of the two real numbers $x$ and $y$ such that $(x+i y)(4-7 i)=3+2 i$
Find the complex number $z$ and write it in the form $a+$ bi if $i(z+1)=3 z-2$
Find the complex number $z$ and write it in the form $a+$ bi if $\frac{2-i}{1+2 i} \sqrt{z}=2-3 i$.
Find the values of the two real numbers $x$ and $y$ such that $(x+i y)^{2}=3-4 i$
a) Find the values of the two real numbers $x$ and $y$ such that $(x+i y)^{2}=-8+6 i$b) Hence, solve the following equation$$z^{2}+(1-i) z+2-2 i=0$$
If $z \in \mathbb{C},$ find all solutions to the equation $z^{3}-27 i=0$.
Given that $z=\frac{1}{2}+2 i$ is a zero of the polynomial $f(x)=4 x^{3}-16 x^{2}+29 x-51$ find the other zeros.
Find a polynomial function with integer coefficients and lowest possible degree that has $\frac{1}{2},-1$ and $3+i \sqrt{2}$ as zeros.
Find a polynomial function with integer coefficients and lowest possible degree that has -2,-2 and $1+i \sqrt{3}$ as zeros.
Given that $z=5+2 i$ is a zero of the polynomial $f(x)=x^{3}-7 x^{2}-x+87,$ find the other zeros.
Given that $z=1-i \sqrt{3}$ is a zero of the polynomial $f(x)=3 x^{3}-4 x^{2}+8 x+8$ find the other zeros.
Let $z \in \mathbb{C} .$ If $\frac{z}{z^{*}}=a+b i,$ show that $|a+b i|=1$
Given that $z=(k+i)^{4}$ where $k$ is a real number, find all values of $k$ such thata) $z$ is a real numberb) $z$ is purely imaginary.
Solve the system of equations.$$\begin{array}{l}i z_{1}+2 z_{2}=3-i \\2 z_{1}+(2+i) z_{2}=7+2 i\end{array}$$
Solve the system of equations.$i z_{1}-(1+i) z_{2}=3$$(2+i) z_{1}+i z_{2}=4$