Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √6x⋅√3x^2

Verified step by step guidance

1. Identify the expression: $\sqrt{6x}\cdot \sqrt{3x^2}$.
2. Apply the product rule for square roots: $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$.
3. Combine the expressions under a single square root: $\sqrt{6x\cdot 3x^2}$.
4. Multiply the terms inside the square root: $6\cdot 3=18$ and $x\cdot x^2=x^3$.
5. Simplify the expression: $\sqrt{18x^3}$.

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.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying algebraic rules to make expressions easier to work with. This includes factoring, distributing, and using properties of exponents and roots. Mastery of simplification techniques is vital for solving equations and performing operations in algebra.