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

Verified step by step guidance

1. Recall the product rule for radicals, which states that for non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
2. Apply the product rule to the expression \(\sqrt{3} \cdot \sqrt{27}\) by combining the radicals under a single square root: \(\sqrt{3 \cdot 27}\).
3. Multiply the numbers inside the radical: \(3 \cdot 27 = 81\), so the expression becomes \(\sqrt{81}\).
4. Recognize that \(\sqrt{81}\) is a perfect square, since \$81 = 9^2$, so it can be simplified further to \(9\).
5. Thus, the original expression \(\sqrt{3} \cdot \sqrt{27}\) simplifies to \(9\) by using the product rule for radicals.

Key Concepts

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.

Simplifying Radicals

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.

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. This rule is useful for rewriting expressions involving division under radicals, though it is not directly applied in this multiplication problem.