Question

Express the following in the form of $a+ib,a,b\in R$

$\frac{4-\sqrt{-9}}{3+\sqrt{-16}}+\frac{3-\sqrt{-25}}{4-\sqrt{-9}}$.

Where, $i=\sqrt{-1}.$ State the values of a and b.

Answer 1

$\begin{array}{c}Expressina+ibform.\\ \frac{4-\sqrt{-9}}{3+\sqrt{-16}}+\frac{3-\sqrt{-25}}{4-\sqrt{-9}}\\ =\frac{4-3i}{3+4i}+\frac{3-5i}{4-3i}\\ =\frac{{\left(4-3i\right)}^2+\left(3-5i\right).\left(3+4i\right)}{\left(3+4i\right)\left(4-3i\right)}\\ =\frac{9\left(4-3i\right)}{\left(24+7i\right)}\times \frac{\left(24-7i\right)}{\left(24-7i\right)}\\ =9\left(\frac{75-100i}{625}\right)\\ =\left(\frac{27}{25}-\frac{36}{25}i\right)\end{array}$

$\therefore \text{The value of a is}\frac{27}{25}\text{and the value of b is}\frac{-36}{25}$