Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Exponents

College Algebra

0. Review of Algebra

Exponents

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1:56 minutes

Problem 113

Textbook Question

Identify the property illustrated in each statement. Assume all variables represent real numbers. 612+615=6(12+15)

Verified step by step guidance

Identify the expression on the left side: \(6 \times 12 + 6 \times 15\).

Notice that both terms on the left side have a common factor of 6.

Factor out the common factor of 6 from the left side: \(6(12) + 6(15) = 6(12 + 15)\).

Recognize that this transformation is an example of the Distributive Property.

The Distributive Property states that \(a(b + c) = ab + ac\), which is what is illustrated here.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distributive Property

The Distributive Property states that for any real numbers a, b, and c, the equation a(b + c) = ab + ac holds true. This property allows us to distribute a multiplication operation over addition, simplifying expressions and solving equations. In the given statement, 6(12 + 15) illustrates this property by showing that multiplying 6 by the sum of 12 and 15 is equivalent to multiplying 6 by each number separately and then adding the results.

Commutative Property of Addition

The Commutative Property of Addition states that the order in which two numbers are added does not affect the sum. In other words, a + b = b + a for any real numbers a and b. This property is implicitly used in the expression 12 + 15, as the sum remains the same regardless of the order in which the numbers are added, allowing for flexibility in rearranging terms.

Associative Property of Addition

The Associative Property of Addition indicates that when adding three or more numbers, the way in which the numbers are grouped does not change the sum. For example, (a + b) + c = a + (b + c). In the context of the given statement, this property is relevant as it allows for the grouping of 12 and 15 within the parentheses, ensuring that the addition can be performed in any order without affecting the final result.