Rationalize the denominator:
The problem asks to simplify the expression \(\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}\) by rationalizing the denominator. Rationalizing a denominator is the process of removing the square root from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by an appropriate expression that results in a rational number in the denominator. Here, we are already given the multiplier \(\frac{\sqrt{3}}{\sqrt{3}}\), which is equal to 1, thus preserving the value of the original fraction \(\frac{5}{\sqrt{3}}\).
To multiply two fractions, we multiply their numerators and their denominators separately:
\[ \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} \]
In the numerator, we have \(5 \cdot \sqrt{3}\), which is simply \(5\sqrt{3}\). In the denominator, we have the product \(\sqrt{3} \cdot \sqrt{3}\). By the definition of a square root, \(\sqrt{x} \cdot \sqrt{x} = x\) for any \(x \ge 0\). Therefore, \(\sqrt{3} \cdot \sqrt{3} = 3\).
Combining the simplified numerator and denominator, we obtain the resulting fraction:
\[ \frac{5\sqrt{3}}{3} \]
This expression is now rationalized as the denominator is the integer 3. No further simplification is possible. The final answer is \(\boxed{\frac{5\sqrt{3}}{3}}\).
Rationalizing denominators is frequently used in geometry and trigonometry. For example, if you are calculating the side lengths of a \(30^\circ-60^\circ-90^\circ\) triangle, you often encounter expressions like \(\frac{1}{\sqrt{3}}\). Rationalizing this to \(\frac{\sqrt{3}}{3}\) makes it much easier to compare lengths, perform further arithmetic, or input the values into formulas. This technique ensures consistency and simplifies calculations in fields ranging from architecture to physics.