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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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.

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