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

Key Concepts

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.

Simplifying Radicals

Simplifying radicals involves reducing expressions that contain square roots to their simplest form. This often 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 x^4/x, allowing for easier calculations and clearer results.