Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √24x^4/√3x

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Rewrite the given expression as a single fraction under one square root: $ \frac{\sqrt{24x^4}}{\sqrt{3x}} = \sqrt{\frac{24x^4}{3x}} $.

Simplify the fraction inside the square root by dividing the coefficients and subtracting the exponents of $x$ (using the property $ \frac{a^m}{a^n} = a^{m-n} $): $ \frac{24x^4}{3x} = 8x^{4-1} = 8x^3 $.

Substitute the simplified fraction back into the square root: $ \sqrt{8x^3} $.

Break the square root into two parts using the property $ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} $: $ \sqrt{8x^3} = \sqrt{8} \cdot \sqrt{x^3} $.

Simplify each part: $ \sqrt{8} = 2\sqrt{2} $ (since $8 = 4 \cdot 2$ and $\sqrt{4} = 2$), and $ \sqrt{x^3} = x^{3/2} = x \cdot \sqrt{x} $. Combine these results to express the simplified form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Simplifying Radicals

Simplifying radicals involves reducing expressions that contain square roots to their simplest form. This process includes factoring out perfect squares from under the radical sign and simplifying the expression accordingly. For example, √(a*b) can be expressed as √a * √b, which is crucial for handling expressions like √(24x^4) and √(3x) in the given problem.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). These properties are vital for simplifying expressions like 24x^4/3x, as they allow for the combination and reduction of terms efficiently.

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