Textbook Question
State the name of the property illustrated: (6 • 3) • 9 = 6 • (3 • 9)
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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 14
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27
Verified step by step guidance1
Identify the expression to simplify, which is \( \sqrt{27} \). Recognize that \( \sqrt{27} \) can be rewritten using the property of square roots as \( \sqrt{27} = \sqrt{9 \times 3} \).
Apply the product rule for square roots, which states that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), to rewrite \( \sqrt{27} \) as \( \sqrt{9} \times \sqrt{3} \).
Simplify the square root of the perfect square: \( \sqrt{9} = 3 \). So the expression becomes \( 3 \times \sqrt{3} \).
Express the simplified form as \( 3\sqrt{3} \), which is the simplified radical form of \( \sqrt{27} \).
Confirm that the variables represent nonnegative real numbers, ensuring the square root expressions are valid and the simplification holds.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule for Radicals
The product rule for radicals states that the square root of a product equals the product of the square roots: √(a * b) = √a * √b. This rule helps simplify expressions by breaking down complex radicals into simpler factors.
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Guided course
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Expanding Radicals
Simplifying Square Roots
Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. For example, √27 = √(9 * 3) = 3√3.
Recommended video:
02:20Imaginary Roots with the Square Root Property
Nonnegative Real Numbers Assumption
Assuming variables represent nonnegative real numbers ensures that square roots are defined as principal (nonnegative) roots. This assumption avoids ambiguity in simplification and allows the use of the product rule without considering complex numbers.
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03:31Introduction to Complex Numbers
Related Practice
Textbook Question
Evaluate each exponential expression in Exercises 1–22.
Textbook Question
State the name of the property illustrated: 3+17 = 17+3
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Textbook Question
In Exercises 11–16, factor by grouping. x3+6x2−2x−12
Textbook Question
In Exercises 15–58, find each product. (x+1)(x2−x+1)
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Textbook Question
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x2−14x+49)/(x2−49)
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