Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √5 /√20

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{\sqrt{5}}{\sqrt{20}}\).
2. Use the quotient rule for radicals, which states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), to rewrite the expression as \(\sqrt{\frac{5}{20}}\).
3. Simplify the fraction inside the radical: \(\frac{5}{20} = \frac{1}{4}\).
4. Rewrite the expression as \(\sqrt{\frac{1}{4}}\).
5. Use the product rule for radicals to separate the square root of the fraction: \(\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}\), then simplify each radical.

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 rule allows simplification by breaking down complex radicals into simpler factors, making it easier to simplify or combine expressions.

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, where b ≠ 0. This rule helps in rewriting expressions involving division under a radical sign into a simpler form.

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares and other factors, then applying the product rule to extract square roots of perfect squares. For example, √20 can be simplified to 2√5 by factoring 20 as 4 * 5.