Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{2 x^{2}} \cdot \sqrt{6 x}$

검증된 단계별 안내

Identify the two expressions being multiplied: $\sqrt{2x^2}$ and $\sqrt{6x}$.

Recall the product rule for square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which allows us to combine the square roots into one.

Apply the product rule: $\sqrt{2x^2} \cdot \sqrt{6x} = \sqrt{(2x^2)(6x)}$.

Multiply the expressions inside the square root: $(2x^2)(6x) = 12x^3$, so the expression becomes $\sqrt{12x^3}$.

Simplify $\sqrt{12x^3}$ by factoring inside the root to extract perfect squares, such as $12 = 4 \times 3$ and $x^3 = x^2 \times x$, then rewrite as $\sqrt{4 \times 3 \times x^2 \times x}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Product Rule for Radicals

The product rule for radicals states that the square root of a product equals the product of the square roots, i.e., √a * √b = √(a*b). This allows simplification by combining the radicands under a single square root before further simplification.

Properties of Exponents

When multiplying expressions with the same base, add their exponents: x^m * x^n = x^(m+n). This property helps simplify terms like x^2 * x by combining the powers into a single exponent.

Simplification of Radicals with Variables

When simplifying radicals involving variables, consider the domain (nonnegative real numbers) to safely apply square root properties. For example, √(x^2) = x if x ≥ 0, ensuring the expression remains valid and simplified.

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 2x2⋅6x\(\sqrt{2x^2}\[\cdot\]\sqrt{6x}\)