In mathematics, it is essential to understand that radicals should not be left in the denominator of a fraction. This practice is known as rationalizing the denominator, which is a straightforward process that allows us to eliminate radicals from the bottom of a fraction. For instance, while expressions like \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{\sqrt{8}} \) can sometimes simplify to perfect squares, there are cases where this is not possible, such as \( \frac{1}{\sqrt{3}} \).

To rationalize the denominator, you multiply both the numerator and the denominator by the radical present in the denominator. For example, to rationalize \( \frac{1}{\sqrt{3}} \), you would multiply by \( \sqrt{3} \) over itself, resulting in:

\[\frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}\]

This transformation is valid because multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \) is equivalent to multiplying by 1, thus preserving the value of the expression. The denominator now becomes a rational number, specifically \( 3 \), while the numerator remains as \( \sqrt{3} \).

To verify the equivalence of the two expressions, you can use a calculator. Inputting \( \frac{1}{\sqrt{3}} \) yields approximately \( 0.57 \), which matches the result of \( \frac{\sqrt{3}}{3} \). This demonstrates that both forms represent the same value, but rationalizing the denominator provides a more conventional representation in mathematics.