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

Verified step by step guidance

1. Recognize that the expression involves the product of two square roots: \( \sqrt{3} \cdot \sqrt{27} \).
2. Apply the product rule for radicals, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
3. Combine the expressions under a single square root: \( \sqrt{3 \cdot 27} \).
4. Calculate the product inside the square root: \( 3 \cdot 27 = 81 \).
5. Rewrite the expression as \( \sqrt{81} \) and simplify if possible.

Key Concepts

Product Rule for Radicals

The product rule for radicals states that the square root of a product is equal to the product of the square roots. In mathematical terms, √a • √b = √(a • b). This rule simplifies the multiplication of square roots, allowing for easier calculations and simplifications in expressions involving radicals.

Quotient Rule for Radicals

The quotient rule for radicals states that the square root of a quotient is equal to the quotient of the square roots. Specifically, √(a/b) = √a / √b. This rule is useful for simplifying expressions where a radical is divided by another radical, making it easier to work with fractions involving square roots.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. This often includes factoring out perfect squares from under the radical sign. For example, √27 can be simplified to √(9 • 3) = √9 • √3 = 3√3, which makes calculations more manageable and clearer.