Textbook Question
Identify the property illustrated in each statement. Assume all variables represent real numbers. 6 • 12 + 6 • 15 = 6(12 + 15)
Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.2.113
Identify the property illustrated in each statement. Assume all variables represent real numbers. (5x) • (1/5x) = 5 ( x • 1/x )
Verified step by step guidance1
Observe the given expression: \((5x) \left( \frac{1}{x} \right) = 5 \left( x \cdot \frac{1}{x} \right)\).
Recognize that the expression shows how multiplication distributes over another operation, specifically how a product can be separated or grouped.
Recall the Distributive Property, which states that for all real numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). However, here we are dealing with multiplication and division, so consider how multiplication interacts with division.
Notice that multiplying \$5x\( by \(\frac{1}{x}\) can be rewritten as \(5\) times the product of \)x$ and \(\frac{1}{x}\), which is exactly what the right side shows: \(5 \left( x \cdot \frac{1}{x} \right)\).
Conclude that this illustrates the Associative Property of Multiplication, which allows us to regroup factors without changing the product, i.e., \((ab)c = a(bc)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distributive Property of Multiplication over Division
This property states that multiplying a product by a fraction is equivalent to multiplying each factor by the fraction separately. In the given expression, (5x) * (1/x) = 5 * (x * 1/x), the multiplication distributes over the division, simplifying the expression.
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Simplification of Fractions
Simplifying fractions involves reducing expressions by canceling common factors in the numerator and denominator. Here, x in the numerator and denominator cancels out, which is essential to understand how the expression simplifies to 5.
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Properties of Real Numbers
Real numbers follow specific algebraic rules such as commutativity, associativity, and distributivity. Recognizing these properties helps in manipulating and simplifying expressions involving variables and constants, as seen in the problem.
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Related Practice
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