Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √3 • √5

Verified step by step guidance

1. Recall the product rule for radicals, which states that for any non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
2. Apply the product rule to the given expression \(\sqrt{3} \cdot \sqrt{5}\) by combining the radicals under a single square root: \(\sqrt{3 \cdot 5}\).
3. Multiply the numbers inside the radical: \(3 \cdot 5 = 15\), so the expression becomes \(\sqrt{15}\).
4. Check if the number inside the radical can be simplified further by factoring out perfect squares. Since 15 has no perfect square factors other than 1, it cannot be simplified further.
5. Therefore, the expression \(\sqrt{3} \cdot \sqrt{5}\) is rewritten as \(\sqrt{15}\) using the product rule for radicals.

Key Concepts

Product Rule for Radicals

The product rule for radicals states that the square root of a product equals the product of the square roots: √a • √b = √(a•b). This allows simplification by combining radicals under a single root, making expressions easier to handle.

Quotient Rule for Radicals

The quotient rule for radicals states that the square root of a quotient equals the quotient of the square roots: √(a/b) = √a / √b. This rule helps in simplifying expressions involving division under radicals.

Simplification of Radicals

Simplifying radicals involves rewriting expressions to their simplest form by factoring out perfect squares or combining terms using product and quotient rules. This process makes expressions clearer and easier to work with in further calculations.