Textbook Question
Simplify each exponential expression in Exercises 23–64.
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 1, Problem 61
Simplify the radical expression. 4∛16 + 5∛2
Verified step by step guidance1
Identify the radical expressions to simplify: \(4\sqrt[3]{16} + 5\sqrt[3]{2}\).
Express the radicands in terms of prime factors or powers to see if they can be simplified: \(16 = 2^4\) and \(2 = 2^1\).
Rewrite the cube roots using the prime factorization: \(\sqrt[3]{16} = \sqrt[3]{2^4} = \sqrt[3]{2^3 \cdot 2} = \sqrt[3]{2^3} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2}\).
Substitute back into the original expression: \(4 \cdot (2 \cdot \sqrt[3]{2}) + 5 \cdot \sqrt[3]{2}\).
Factor out the common cube root term \(\sqrt[3]{2}\): \((4 \cdot 2 + 5) \cdot \sqrt[3]{2}\) and simplify inside the parentheses.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simplifying Radicals
Simplifying radicals involves rewriting the expression so that the radicand (the number inside the root) has no perfect powers corresponding to the root's index. For cube roots, this means factoring out perfect cubes to simplify the expression. This process makes the radical easier to work with or combine.
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Properties of Cube Roots
Cube roots are the inverse operation of cubing a number. The cube root of a product equals the product of the cube roots, i.e., ∛(a * b) = ∛a * ∛b. This property allows breaking down complex radicands into simpler factors to simplify or combine terms.
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Combining Like Radicals
Like radicals have the same index and radicand after simplification. Only like radicals can be added or subtracted by combining their coefficients. Recognizing and simplifying radicals to their simplest form is essential before combining them.
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Related Practice
Textbook Question
Evaluate each algebraic expression for x = 2 and y= -5. |x+y|
Textbook Question
In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (7x4y2−5x2y2+3xy)+(−18x4y2−6x2y2−xy)
Textbook Question
In Exercises 33–68, add or subtract as indicated. 4/(x2+6x+9) + 4/(x+3)
Textbook Question
Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number.
Textbook Question
Factor using the formula for the sum or difference of two cubes.