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

Verified step by step guidance

1. Identify the expression under the radical: \( \frac{7}{16} \).
2. Apply the quotient rule for radicals: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
3. Rewrite the expression as \( \frac{\sqrt{7}}{\sqrt{16}} \).
4. Recognize that \( \sqrt{16} \) is a perfect square and simplify it.
5. Express the final result as \( \frac{\sqrt{7}}{4} \).

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 allows for the simplification of expressions involving multiplication under a square root, making it easier to manipulate and solve radical expressions.

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 essential for simplifying expressions that involve division under a square root, allowing for clearer and more manageable forms of radical expressions.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form, which often includes factoring out perfect squares. For example, √(16/7) can be simplified to 4/√7. This process is crucial for solving equations and performing operations with radicals, ensuring that the expressions are as straightforward as possible.