Question

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

$\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}\cdot \frac{x}{y}=\frac{c}{d}$

Short answer

Rationalizing the given number, we get$\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}$.

Step-by-step solution

Step 1. Rationalizing the denominator.

It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator. If the denominator is of the form$\sqrt{a}$, we multiply numerator and denominator by$\sqrt{a}$. In doing this we multiply the given quantity by 1, so we do not change its value. For instance,

$\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{a}}\cdot \frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}$

Note that the denominator in the last fraction contains no radical.

A fractional expression whose denominator contains no radicals is said to be in standard form.

Step 2. Evaluate the given value.

Given$\frac{1}{\sqrt{3}}$

We will rationalize the denominator since it has radical.

Step 3. Rationalize the given value.

Given$\frac{1}{\sqrt{3}}$

To rationalize the denominator, we multiply and divide the given number by$\sqrt{3}$

So, we have

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