Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √500x³/√10x^-1

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Step 1: Recall the quotient rule for radicals, which states that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Apply this rule to the given expression \( \frac{\sqrt{500x^3}}{\sqrt{10x^{-1}}} \). This simplifies to \( \sqrt{\frac{500x^3}{10x^{-1}}} \).

Step 2: Simplify the fraction inside the radical. Divide \( 500 \) by \( 10 \), which gives \( 50 \). For the variable \( x \), use the property of exponents: \( x^3 \div x^{-1} = x^{3 - (-1)} = x^{3 + 1} = x^4 \). The fraction becomes \( \frac{50x^4}{1} \), or simply \( 50x^4 \).

Step 3: Rewrite the expression as \( \sqrt{50x^4} \). Now, simplify the radical by breaking it into two parts: \( \sqrt{50} \cdot \sqrt{x^4} \).

Step 4: Simplify \( \sqrt{x^4} \). Since \( x^4 \) is a perfect square, \( \sqrt{x^4} = x^2 \). For \( \sqrt{50} \), factor \( 50 \) into \( 25 \cdot 2 \), where \( 25 \) is a perfect square. Thus, \( \sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \).

Step 5: Combine the simplified parts. The expression becomes \( 5x^2\sqrt{2} \). This is the simplified form of the original expression.

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주요 개념

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

Quotient Rule

The quotient rule is a fundamental principle in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be found using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. Understanding this rule is essential for simplifying expressions involving division of functions.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In the given expression, √500x^3 and √10x^−1 are radical forms that can be simplified by applying properties of exponents and radicals. Recognizing how to manipulate these expressions is crucial for simplifying the overall expression effectively.

Properties of Exponents

Properties of exponents are rules that govern how to handle mathematical expressions involving powers. Key properties include the product of powers, quotient of powers, and power of a power. For example, when dividing like bases, you subtract the exponents. Mastery of these properties is vital for simplifying expressions that include variables raised to powers, especially in the context of the given problem.

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √500x3/√10x-1