Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √48x³/√3x

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Rewrite the given expression as a single fraction under one square root: $ \frac{\sqrt{48x^3}}{\sqrt{3x}} = \sqrt{\frac{48x^3}{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{48x^3}{3x} = 16x^{3-1} = 16x^2 $.

Substitute the simplified fraction back under the square root: $ \sqrt{16x^2} $.

Use the property of square roots $ \sqrt{a^2} = a $ (for $a > 0$) to simplify $ \sqrt{16x^2} $ into $ 4x $.

The simplified expression is $ 4x $.

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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.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can often be simplified by applying properties of exponents. For example, √a = a^(1/2) and √(a/b) = √a/√b. Recognizing how to manipulate these expressions is crucial for simplifying the given expression involving square roots.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, which can include combining like terms, factoring, and reducing fractions. In the context of the given expression, this means applying the properties of radicals and the quotient rule to rewrite the expression in a simpler, more manageable form. Mastery of simplification techniques is vital for solving algebraic problems efficiently.

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