Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √3 • √27

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Recall the product rule for radicals, which states that for non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

Apply the product rule to the expression \(\sqrt{3} \cdot \sqrt{27}\) by combining the radicals under a single square root: \(\sqrt{3 \cdot 27}\).

Multiply the numbers inside the radical: \(3 \cdot 27 = 81\), so the expression becomes \(\sqrt{81}\).

Recognize that \(\sqrt{81}\) is a perfect square, since \(81 = 9^2\), so it can be simplified further to \(9\).

Thus, the original expression \(\sqrt{3} \cdot \sqrt{27}\) simplifies to \(9\) by using the product 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 combining under a single radical, making it easier to evaluate or simplify expressions.

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Simplifying radicals involves expressing the number under the root as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. This process helps in reducing the expression to its simplest form.

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The quotient rule for radicals states that the square root of a quotient equals the quotient of the square roots: √(a/b) = √a / √b. This rule is useful for rewriting expressions involving division under radicals, though it is not directly applied in this multiplication problem.

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