EXERCISE 5.1

If the complex number $\left[\left(\frac{1}{3}+i\frac{7}{3}\right)+\left(4+i\frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right)$ is expressed in the form of $a+ib$, then find $a+b$.  [write the answer in the form of a fraction].

If the complex number ${\left(1-i\right)}^4$ is expressed in the form of $a+ib$, then find $a+b$.

Express the complex number ${\left(\frac{1}{3}+3i\right)}^3$in the form $a+ib$, and write the value of $b.$

Express the complex number ${\left(-2-\frac{1}{3}i\right)}^3$in the form $a+ib$.

If the multiplicative inverse of the complex number $4-3i$ is $\frac{4+3i}{k}$, then $k$ is

If the multiplicative inverse of the complex number $\sqrt{5}+3i$ is $\frac{\sqrt{5}}{14}-i\frac{k}{14}$, then find value of $k$.

Find the multiplicative inverse of the complex number $-i$.

If the following expression in the form of $a+ib:$$\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$, then $a=$