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

Verified step by step guidance

1. Recognize that the expression involves the product of two square roots: \( \sqrt{3} \cdot \sqrt{5} \).
2. Apply the product rule for radicals, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
3. Substitute the values into the product rule: \( \sqrt{3 \cdot 5} \).
4. Simplify the expression inside the square root: \( \sqrt{15} \).
5. Conclude that the expression \( \sqrt{3} \cdot \sqrt{5} \) simplifies to \( \sqrt{15} \).

Key Concepts

Product Rule for Radicals

The product rule for radicals states that the square root of a product is equal to the product of the square roots. In mathematical terms, √a • √b = √(a • b). This rule simplifies the multiplication of square roots, allowing for easier calculations and simplifications in expressions involving radicals.

Quotient Rule for Radicals

The quotient rule for radicals indicates that the square root of a quotient is equal to the quotient of the square roots. Formally, this is expressed as √(a/b) = √a / √b. This rule is useful for simplifying expressions where a radical is divided by another radical, making it easier to work with fractions involving square roots.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form, which often includes factoring out perfect squares. For example, √(a^2 • b) can be simplified to a√b. This process is essential for making calculations more manageable and for presenting answers in a standard format.