Domanda del libro di testo
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 1, Problema 16
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
Guida verificata passo dopo passo1
Identify the function as a product of two parts: \(\sqrt{125}\) and \(x^2\).
Rewrite the square root and constants in simpler radical or exponential form: \(\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}\).
Express the entire function as \(5\sqrt{5} \cdot x^2\) to clearly see the product of two functions: \(f(x) = 5\sqrt{5}\) and \(g(x) = x^2\).
Apply the product rule for derivatives, which states: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\).
Since \(f(x) = 5\sqrt{5}\) is a constant, its derivative \(f'(x) = 0\). Then find \(g'(x)\) by differentiating \(x^2\) to get \$2x$. Substitute these into the product rule formula to write the derivative expression.
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Product Rule for Radicals
The product rule for radicals states that the square root of a product equals the product of the square roots: √(ab) = √a × √b. This rule allows simplification of expressions by separating factors under the radical into simpler parts.
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Expanding Radicals
Simplifying Square Roots
Simplifying square roots involves factoring the radicand into perfect squares and other factors, then taking the square root of perfect squares outside the radical. For example, √125 can be simplified to 5√5 since 125 = 25 × 5.
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02:20Imaginary Roots with the Square Root Property
Properties of Exponents with Radicals
Variables under radicals can be expressed using fractional exponents, such as √(x^2) = x^(2/2) = x. When variables represent nonnegative real numbers, the square root of x squared simplifies directly to x, ensuring the expression remains valid.
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04:06Rational Exponents
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