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

Verified step by step guidance

1. Step 1: Recognize that the expression involves the product of two square roots: $\sqrt{5}\times \sqrt{20}$.
2. Step 2: Use the product rule for radicals, which states that $\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}$.
3. Step 3: Multiply the numbers under the radicals: $5\times 20=100$.
4. Step 4: Apply the product rule: $\sqrt{5}\times \sqrt{20}=\sqrt{100}$.
5. Step 5: Simplify $\sqrt{100}$ to its simplest form.

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. Formally, √(a / b) = √a / √b. This principle is useful for simplifying expressions where a radical is divided by another number or radical, making calculations easier.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form, which often includes factoring out perfect squares. For example, √20 can be simplified to √(4 * 5) = √4 * √5 = 2√5. This process is essential for making calculations clearer and more straightforward.