Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √10x⋅√ 8x

Verified step by step guidance

1. Recognize that the expression involves the product of two square roots: \( \sqrt{10x} \cdot \sqrt{8x} \).
2. Apply the product rule for square roots, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
3. Combine the expressions under a single square root: \( \sqrt{(10x) \cdot (8x)} \).
4. Multiply the terms inside the square root: \( \sqrt{80x^2} \).
5. Simplify the square root by factoring out perfect squares: \( \sqrt{80} \cdot \sqrt{x^2} \).

Key Concepts

Product Rule of Exponents

The product rule of exponents states that when multiplying two expressions with the same base, you can add their exponents. For example, a^m * a^n = a^(m+n). This rule is essential for simplifying expressions involving powers, particularly when dealing with variables and constants in algebra.

Square Roots

A square root of a number x is a value that, when multiplied by itself, gives x. The square root is denoted as √x. Understanding how to manipulate square roots, including the property that √a * √b = √(a*b), is crucial for simplifying expressions that involve square roots, especially in algebraic contexts.

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 taking square roots yield real results. In the context of the given expression, it implies that the variables involved cannot take negative values, which affects the simplification process.