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The process of simplifying fractions involves reducing the fraction to its simplest form. A simplified fraction has the smallest possible numerator and denominator that represent the same value. To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

In the given exercise, once the denominators have been simplified (i.e., multiplying the square roots of the same number, resulting in the number itself, such as \(\sqrt{5} \cdot \sqrt{5} = 5\)), the final fraction simplifies. Here, because the numerator \(\sqrt{15}\) and denominator 5 have no common divisor besides 1, the fraction is already in simplest form as \(\frac{\sqrt{15}}{5}\). This process helps in understanding equivalent fractions and ensures accuracy in calculations.

Multiplying fractions is straightforward once you remember that all you need to do is multiply straight across the numerators and denominators. To multiply fractions:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For example, in the exercise: \(\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\), multiply the top numbers (\(\sqrt{3} \) and \(\sqrt{5} \)) and the bottom numbers (\(\sqrt{5} \) and \(\sqrt{5} \)), giving you \(\frac{\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\). This reflects the essential rule of multiplying the terms directly. It’s crucial to simplify the result whenever possible for precision.

Square roots have interesting properties that come in handy, especially when working with fractions. Understanding these properties is key:

• The square root of a product is the product of the square roots: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
• The square root of a square of a number returns the number itself: \(\sqrt{a^2} = a\).

Applying these rules in the context of the exercise means:

• For the numerator \(\sqrt{3} \cdot \sqrt{5} = \sqrt{15}\)
• For the denominator \(\sqrt{5} \cdot \sqrt{5} = 5\), because the square of \(\sqrt{5}\) simplifies to the original number under the square root.

Knowing these principles makes working with roots in fractional form simpler and more efficient, ensuring you can accurately simplify expressions.