Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √2x^2⋅√6x

Verified step by step guidance

1. Identify the expression: $\sqrt{2x^2}\cdot \sqrt{6x}$.
2. Use the property of square roots: $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$.
3. Combine the expressions under a single square root: $\sqrt{(2x^2)\cdot (6x)}$.
4. Multiply the terms inside the square root: $\sqrt{12x^3}$.
5. Simplify the expression by factoring out perfect squares: $\sqrt{4\cdot 3x^3}=\sqrt{4}\cdot \sqrt{3x^3}$.

Key Concepts

Product Rule

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

Square Roots

Square roots are a mathematical operation that finds a number which, when multiplied by itself, gives the original number. The square root of a product can be expressed as the product of the square roots of the individual factors, i.e., √(a*b) = √a * √b. This property is crucial for simplifying expressions that involve square roots.

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. Understanding this helps in correctly applying algebraic rules and simplifying expressions without introducing extraneous solutions.