Express each of the complex number given in the form of $a+ib$
1. $(5i)(-\frac{3}{5}i)$
= $-5*\frac{3}{5}*i*i$
= $-3*i^{2}$
= $-3*(-1)$ [$i^{2}=-1$ ]
= $3$
2. $i^{9}+i^{19}$
= $i^{4*2+1}+i^{4*4+3}$
= $(i^{4})^{2}.i+(i^{4})^{4}.i^{3}$  [$i^{4}=1$ , $i^{3}=-i$ ]
= $1.i+1.(-i)$
= 0
3. $i^{-39}$
= $(i^{4})^{-9}.i^{-3}$
= $(1)^{-9}.\frac{1}{i^{3}}$
= $\frac{1}{-i}$
= $-\frac{1}{i}*\frac{i}{i}$
= $-\frac{i}{i^{2}}$
= $-\frac{i}{-1}$
= $i$
4.  $3(7+i7)+i(7+i7)$
= $21+i21+i7+i^{2}7$
= $21+i28-7$
= $14+i28$
5. $(1-i)-(-1+i6)$
= $1-i+1-i6$
= $2-i7$
6. $\left ( \frac{1}{5}+i\frac{2}{5} \right )-\left ( 4+i\frac{5}{2} \right )$
= $\frac{1}{5}+i\frac{2}{5}-4-i\frac{5}{2}$
= $(\frac{1}{5}-4)+i(\frac{2}{5}-\frac{5}{2})$
= $-\frac{19}{5}+i\frac{-21}{10}$
= $-\frac{19}{5}-i\frac{21}{10}$
7. $\left [ (\frac{1}{3}+i\frac{7}{3})+(4+i\frac{1}{3}) \right ]$ $-(\frac{-4}{3}+i)$
= $\left [ (\frac{1}{3}+4)+i(\frac{7}{3}+\frac{1}{3}) \right ]$ $-(\frac{-4}{3}+i)$
= $\frac{13}{3}+i\frac{8}{3}+\frac{4}{3}-i$
= $(\frac{13}{3}+\frac{4}{3})+i(\frac{8}{3}-1)$
= $\frac{17}{3}+i\frac{5}{3}$
8. $(1-i)^{4}$
= $[(1-i)^{2}]^{2}$
= $(1+i^{2}-2i)^{2}$
= $(1-1-2i)^{2}$
= $(-2i)^{2}$
= $4i^{2}$
= $-4$
9. $(\frac{1}{3}+3i)^{3}$
= $(\frac{1}{3})^{3}+3.(\frac{1}{3})^{2}.(3i)$ $+3.(\frac{1}{3}).(3i)^{2}+(3i)^{3}$
= $\frac{1}{27}+i+9i^{2}+27i^{3}$
= $\frac{1}{27}+i-9-27i$
= $-\frac{242}{27}-i26$
10.  $(-2-\frac{1}{3}i)^{3}$
= $(-1)^{3}(2+\frac{1}{3}i)^{3}$
= $-[2^{3}+3.(2)^{2}.\frac{1}{3}i$ $+3.2.(\frac{1}{3}i)^{2}+(\frac{1}{3}i)^{3}]$
= $-[8+i4+\frac{2}{3}i^{2}+\frac{1}{27}i^{3}]$
= $-[8+i4-\frac{2}{3}-i\frac{1}{27}]$
= $-[\frac{22}{3}+i\frac{107}{27}]$
= $-\frac{22}{3}-i\frac{107}{27}$
Find the multiplicative inverse of each of the complex numbers given.
11. $4-3i$
$\bar{z}=4+3i$
$|z|^{2}=4^{2}+3^{2}$
= $16+9=25$
$|z|^{2}=25$
Multiplicative inverse, $z^{-1}$ $\large =\frac{\bar{z}}{|z|^{2}}$
= $\frac{4+3i}{25}$
$z^{-1}=\frac{4}{25}+i\frac{3}{25}$
12. $\sqrt{5}+3i$
$\bar{z}=\sqrt{5}-3i$
$|z|^{2}=(\sqrt{5})^{2}+3^{2}$
= $5+9=14$
$|z|^{2}=14$
Multiplicative inverse, $z^{-1}$ $\large =\frac{\bar{z}}{|z|^{2}}$
= $\frac{\sqrt{5}-3i}{14}$
$z^{-1}=\frac{\sqrt{5}}{14}-i\frac{3}{14}$
13. $-i$
$\bar{z}=i$
$|z|^{2}=1^{2}$
$|z|^{2}=1$
Multiplicative inverse, $z^{-1}$ $\large =\frac{\bar{z}}{|z|^{2}}$
= $\frac{i}{1}$
$z^{-1}=i$
14. Express the following expression in the form of $a+ib$
$\large \frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+i\sqrt{2})-(\sqrt{3}-i\sqrt{2})}$
= $\large \frac{(3)^{2}-(i\sqrt{5})^{2}}{\sqrt{3}+i\sqrt{2}-\sqrt{3}+i\sqrt{2}}$ $ [(a+b)(a-b)=a^{2}-b^{2}]$
= $\large \frac{9-i^{2}5}{i2\sqrt{2}}$
= $\large \frac{9+5}{i2\sqrt{2}}$
= $\large \frac{14}{i2\sqrt{2}}$
= $\large \frac{7}{i\sqrt{2}}$
= $\large \frac{7}{i\sqrt{2}}*\frac{i\sqrt{2}}{i\sqrt{2}}$
= $\large \frac{i7\sqrt{2}}{i^{2}(\sqrt{2})^{2}}$
= $\large \frac{i7\sqrt{2}}{-2}$
= $\large -i\frac{7\sqrt{2}}{2}$