Textbook Question
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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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
State the name of the property illustrated: (6 • 3) • 9 = 6 • (3 • 9)
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Identify the operation involved in the expression. Here, the operation is multiplication, denoted by the symbol \( \cdot \).
Observe how the grouping of the numbers changes in the equation: the first grouping is \( (6 \cdot 3) \cdot 9 \) and the second grouping is \( 6 \cdot (3 \cdot 9) \).
Recall the property that states that when multiplying three or more numbers, the way in which the numbers are grouped does not affect the product. This property is known as the Associative Property of Multiplication.
Write the general form of the Associative Property of Multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \), where \(a\), \(b\), and \(c\) are any real numbers.
Conclude that the given equation \( (6 \cdot 3) \cdot 9 = 6 \cdot (3 \cdot 9) \) illustrates the Associative Property of Multiplication.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Associative Property of Multiplication
This property states that when multiplying three or more numbers, the way in which the numbers are grouped does not affect the product. In other words, (a • b) • c = a • (b • c). The example (6 • 3) • 9 = 6 • (3 • 9) illustrates this property clearly.
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Change of Base Property
Multiplication Operation
Multiplication is a basic arithmetic operation that combines equal groups. Understanding how multiplication works is essential to grasp properties like associativity, as it involves combining numbers in different groupings without changing the result.
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Guided course
8:38Performing Row Operations on Matrices
Properties of Real Numbers
Real numbers follow specific properties such as commutative, associative, and distributive properties. Recognizing these properties helps in simplifying expressions and solving equations efficiently, as shown in the given multiplication example.
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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 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (5x2−7x−8)+(2x2−3x+7)−(x2−4x−3)
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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