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

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Identify the expression to simplify: \(\frac{\sqrt{5}}{\sqrt{20}}\).

Use the quotient rule for radicals, which states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), to rewrite the expression as \(\sqrt{\frac{5}{20}}\).

Simplify the fraction inside the radical: \(\frac{5}{20} = \frac{1}{4}\).

Rewrite the expression as \(\sqrt{\frac{1}{4}}\).

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.

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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.

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Introduction to Dot Product

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.

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Quotients of Complex Numbers in Polar 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.

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