Chapter A.7, Problem 92AYU

To determine

To Explain:The method used to rationalize the denominator of a radical expression and the method used to write the quotient of two complex numbers.

Explanation of Solution

Given: Write a brief paragraph that compares the method used to rationalize the denominator of a radical expression and the method used to write the quotient of two complex numbers in standard form.

The method used to rationalize the denominator of a radical expression

For rationalizing the denominator of a radical expression, multiply and divide by the conjugate of the denominator.

For example:

  $\begin{array}{l}\frac{a+\sqrt{b}}{c+\sqrt{d}}\\ rationalizing\\ =\frac{a+\sqrt{b}}{c+\sqrt{d}}\cdot \frac{c-\sqrt{d}}{c-\sqrt{d}}\\ =\frac{(a+\sqrt{b})(c-\sqrt{d})}{(c+\sqrt{d})(c+\sqrt{d})}\\ =\frac{(a+\sqrt{b})(c-\sqrt{d})}{c^2-d}\end{array}$

The method used to write the quotient of two complex numbers in standard form.

For writing the quotient of two complex numbers in standard form we again multiply and divide by the conjugate of the denominator of the complex number.

For example:

  $\frac{a+bi}{c+di}$

The conjugate of the denominator is $c-di$

Then

  $\begin{array}{l}\frac{a+bi}{c+di}\\ =\frac{a+bi}{c+di}\cdot \frac{c-di}{c-di}=\frac{(a+bi)(c-di)}{c^2+d}\end{array}$