Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27

Key Concepts

Product Rule

The product rule is a fundamental principle in algebra that states when multiplying two expressions, you can simplify the product by applying the rule to each factor. Specifically, if you have two functions, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for simplifying expressions involving products of variables or functions.

Square Root Simplification

Simplifying square roots involves breaking down a square root into its prime factors to express it in a simpler form. For example, √27 can be simplified by recognizing that 27 = 9 × 3, leading to √27 = √(9 × 3) = √9 × √3 = 3√3. This concept is crucial for handling expressions that include square roots, especially when combined with other algebraic operations.

Nonnegative Real Numbers

Nonnegative real numbers are all real numbers that are either positive or zero. This concept is important in algebra as it restricts the domain of variables, ensuring that operations like square roots yield real results. When simplifying expressions, understanding that variables represent nonnegative values helps avoid complications that arise from negative inputs, particularly in square root calculations.