Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{6 x} \cdot \sqrt{3 x^{2}}$

검증된 단계별 안내

Identify the two functions being multiplied: here, the first function is $\sqrt{6x}$ and the second function is $\sqrt{3x^2}$.

Recall that the product rule for derivatives states: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$, but since the problem asks to simplify the product, focus on simplifying the expression $\sqrt{6x} \cdot \sqrt{3x^2}$ first.

Use the property of square roots that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ to combine the two square roots into one: $\sqrt{6x} \cdot \sqrt{3x^2} = \sqrt{(6x)(3x^2)}$.

Multiply inside the square root: $(6x)(3x^2) = 18x^3$, so the expression becomes $\sqrt{18x^3}$.

Simplify $\sqrt{18x^3}$ by factoring inside the root to extract perfect squares: write \$18x^3$ as $9 \cdot 2 \cdot x^2 \cdot x$, then use $\sqrt{a^2} = a$ to simplify.

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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 rule allows simplification by combining radicals under a single square root when multiplying.

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 and x^2 inside or outside radicals.

Simplifying Radicals with Variables

To simplify radicals containing variables, express variables with exponents inside the root and apply exponent rules. For nonnegative variables, √(x^2) simplifies to x, ensuring the expression remains valid.

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 6x⋅3x2\(\sqrt{6x}\[\cdot\]\sqrt{3x^2}\)