Chapter 6.8, Problem 40WE

To determine

To find the reciprocal of the given complex number.

Answer to Problem 40WE

  $\frac{1}{-\sqrt{5}-i\sqrt{2}}=-\frac{\sqrt{5}}{7}+\frac{\sqrt{2}}{7}i$

Explanation of Solution

Given information:

Given complex number $-\sqrt{5}-i\sqrt{2}$

We have to simplify $\frac{1}{-\sqrt{5}-i\sqrt{2}}$

We know that $i^2=-1$ and $i=\sqrt{-1}$

Here we shall also use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$

To solve this we have to multiply numerator and denominator by the conjugate of the denominator.

Hence we get,

  $\begin{array}{c}\frac{1}{-\sqrt{5}-i\sqrt{2}}\times \frac{-\sqrt{5}+i\sqrt{2}}{-\sqrt{5}+i\sqrt{2}}=\frac{-\sqrt{5}+i\sqrt{2}}{{\left(-\sqrt{5}\right)}^2-{\left(\sqrt{2}i\right)}^2}\\ =\frac{-\sqrt{5}+i\sqrt{2}}{5-2i^2}\\ =\frac{-\sqrt{5}+i\sqrt{2}}{7}\\ =-\frac{\sqrt{5}}{7}+\frac{\sqrt{2}}{7}i\end{array}$