Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √4⁄50

Verified step by step guidance

1. Start by simplifying the expression \( \frac{\sqrt{4}}{\sqrt{50}} \).
2. Recognize that \( \sqrt{4} \) can be simplified to 2, since 4 is a perfect square.
3. For \( \sqrt{50} \), factor 50 into its prime factors: 50 = 2 \times 5^2.
4. Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to rewrite \( \sqrt{50} \) as \( \sqrt{2} \times \sqrt{5^2} \).
5. Simplify \( \sqrt{5^2} \) to 5, and rewrite the expression as \( \frac{2}{5\sqrt{2}} \).

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 number or radical, making calculations easier.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the square root to its simplest form. This often includes factoring out perfect squares and applying the product and quotient rules. For example, √(4/50) can be simplified by first applying the quotient rule and then simplifying the resulting radicals to achieve a more concise expression.