Chapter 1.2, Problem 5E

Explain how we rationalize a denominator, then complete the following steps to rationalize $\frac{1}{\sqrt{3}}$:

$\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}\cdot \frac{\boxed{}}{\boxed{}}=\frac{\boxed{}}{\boxed{}}$

To determine

To explain: The rule of rationalizing a denominator and to complete the steps to rationalize the expression $\frac{1}{\sqrt{3}}$.

Answer to Problem 5E

The value of expression $\frac{1}{\sqrt{3}}$ in rationalized form is $\underset{\_}{\frac{\sqrt{3}}{3}}$.

Explanation of Solution

The given radical expression is $\frac{1}{\sqrt{3}}$.

Consider an expression $\frac{1}{\sqrt{a}}$.

In order to rationalize the denominator of the given expression, the numerator and denominator of the given expression is multiplied with the value given in the denominator. That is,

$\begin{array}{c}\frac{1}{\sqrt{a}}=\frac{1\cdot \sqrt{a}}{\sqrt{a}\cdot \sqrt{a}}\\ =\frac{\sqrt{a}}{a}\end{array}$

Thus, the rationalized form of the expression $\frac{1}{\sqrt{a}}$ is $\frac{\sqrt{a}}{a}$.

Use the above method and complete the rationalization of the expression $\frac{1}{\sqrt{3}}$ as shown below.

$\begin{array}{c}\frac{1}{\sqrt{3}}=\frac{1\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}}\\ =\frac{\sqrt{3}}{3}\end{array}$

Thus, the value after rationalization is $\frac{\sqrt{3}}{3}$.