Chapter A.7, Problem 92AYU

To determine

To explain: The method which compares the method used to rationalize the denominator of a radical expression and a complex number in standard form.

Explanation of Solution

Given information:

The method of rationalize the denominator.

Calculation:

The method used to rationalize the denominator of a radical expression is,

  $\begin{array}{c}\frac{a+\sqrt{b}}{c+\sqrt{d}}=\frac{a+\sqrt{b}}{c+\sqrt{d}}\frac{\left(c-\sqrt{d}\right)}{\left(c-\sqrt{d}\right)}\\ =\frac{ac-a\sqrt{d}+c\sqrt{b}-\sqrt{bd}}{c^2-d}\end{array}$

The method used to write complex number in standard form is,

  $\begin{array}{c}\frac{z_1}{z_2}=\frac{\left(a+bi\right)}{\left(c+di\right)}\\ =\frac{\left(a+bi\right)}{\left(c+di\right)}\frac{\left(c-di\right)}{\left(c-di\right)}\\ =\frac{ac+ibc-iad+bd}{c^2+d^2}\\ =\frac{\left(ac+bd\right)+\left(bc-ad\right)i}{c^2+d^2}\end{array}$