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

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Identify the expression given: \(\sqrt{\frac{4}{50}}\).

Recall the quotient rule for radicals, which states that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) for positive numbers \(a\) and \(b\).

Apply the quotient rule to rewrite the expression as \(\frac{\sqrt{4}}{\sqrt{50}}\).

Simplify the numerator and denominator separately: \(\sqrt{4}\) and \(\sqrt{50}\). Use the product rule for radicals if needed, where \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\).

Express the simplified form by breaking down \(\sqrt{50}\) into \(\sqrt{25 \times 2}\) and simplify further to get the expression in simplest radical form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product Rule for Radicals

The product rule states that the square root of a product equals the product of the square roots: √(a * b) = √a * √b. This allows you to simplify or rewrite expressions by separating factors under a single radical into individual radicals.

Quotient Rule for Radicals

The quotient rule 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 inside a radical as a fraction of two separate radicals.

Simplifying Radicals

Simplifying radicals involves factoring numbers inside the root to extract perfect squares, reducing the expression to its simplest form. For example, √50 can be rewritten as √(25 * 2) = 5√2, making the expression easier to work with.

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