Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √7⁄16

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Identify the expression given: \(\frac{\sqrt{7}}{16}\). Notice that the radical is only in the numerator.

Recall the product rule for radicals: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), and the quotient rule for radicals: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).

Since the denominator is not under a radical, consider rewriting the denominator as \(\sqrt{16^2}\) to apply the quotient rule if needed, or leave it as is because \(16\) is a perfect square.

Express the denominator as \(\sqrt{16^2} = 16\), then rewrite the original expression as \(\frac{\sqrt{7}}{\sqrt{16^2}}\).

Apply the quotient rule for radicals: \(\frac{\sqrt{7}}{\sqrt{16^2}} = \sqrt{\frac{7}{16^2}}\). This rewrites the expression using the quotient rule for radicals.

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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 allows simplification by separating factors inside a radical into individual roots, making complex expressions easier to handle.

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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, provided b ≠ 0. This rule helps rewrite expressions involving radicals in numerator and denominator separately for simplification.

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Simplifying Radicals

Simplifying radicals involves reducing the expression under the root to its simplest form by factoring out perfect squares or cubes. This process makes expressions easier to interpret and compare, often by rewriting radicals as simpler fractions or integers.

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