Use the product and quotient rules for radicals to rewrite each expression. See Example 4. 30√10 5√2

Verified step by step guidance

1. Identify the expression to be simplified: \( \frac{30\sqrt{10}}{5\sqrt{2}} \).
2. Apply the quotient rule for radicals: \( \frac{a\sqrt{b}}{c\sqrt{d}} = \frac{a}{c} \cdot \sqrt{\frac{b}{d}} \).
3. Simplify the coefficients: \( \frac{30}{5} = 6 \).
4. Simplify the radical part: \( \sqrt{\frac{10}{2}} = \sqrt{5} \).
5. Combine the simplified parts: \( 6\sqrt{5} \).

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 allows for the simplification of expressions involving square roots by breaking them down into more manageable parts.

Quotient Rule for Radicals

The quotient rule for radicals states that the square root of a quotient is equal to the quotient of the square roots. Specifically, √(a / b) = √a / √b. This principle is useful for simplifying expressions where a radical is divided by another radical, making calculations easier.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the radical sign to its simplest form. This can include factoring out perfect squares or other factors that can be simplified. Mastery of this concept is essential for effectively applying the product and quotient rules to rewrite expressions involving radicals.