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Chapter 5: Problem 84

State the associative property of multiplication and give an example.

Short Answer

Expert verified
The associative property of multiplication states that when multiplying three or more numbers, the way the numbers are grouped does not change the result. As an example, \( (2*3)*4 = 2*(3*4) \).

Step by step solution

01

State the Associative Property of Multiplication

The associative property of multiplication states that when multiplying three or more numbers, the way the numbers are grouped does not change the result. Mathematically, this can be expressed as \( (ab)c = a(bc) \) for all a, b, c.
02

Provide an Example

As an example, let's take three numbers, say, 2, 3, and 4. According to the associative property, we can group 2 and 3 together first or 3 and 4 together first without changing the result. So, \( (2*3)*4 = 2*(3*4) \). Calculating both sides, we find it to be true as \( 6*4 = 2*12 \), which equals 24.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Multiplication
When we discuss the properties of multiplication, we're referring to the rules that dictate how multiplication works in various mathematical settings. These properties make calculations easier and more intuitive. One key property is the **associative property of multiplication**.
  • **Associative Property**: This states that how numbers are grouped in a multiplication operation does not affect the product. For example, with numbers 2, 3, and 4, you can group them as \((2 \times 3) \times 4\) or \(2 \times (3 \times 4)\), and the result will be the same, which is 24.
  • **Commutative Property**: This states that the order of multiplying numbers does not change the product, such as \(a \times b = b \times a\).
  • **Distributive Property**: This combines multiplication and addition, allowing you to multiply a sum by multiplying each addend separately and then adding the products.
Understanding these properties can greatly simplify problem-solving and enhance the ability to perform calculations quickly and accurately.
Mathematical Properties
Mathematical properties are the foundational rules that define various operations, offering consistency and reliability in calculations. Having a firm grasp of these properties allows mathematicians and students alike to solve complex problems more efficiently.
  • **Associative Property of Multiplication**: As previously noted, this property helps in regrouping numbers, facilitating easier computation particularly in strings of operations.
  • **Identity Property**: This property states that any number multiplied by one remains unchanged, which means \(a \times 1 = a\).
  • **Zero Property**: Multiplying any number by zero results in zero, \(a \times 0 = 0\). This is crucial in simplifying expressions and solving equations.
These properties are integral to understanding mathematical concepts and performing algebraic operations.
Arithmetic Concepts
Arithmetic concepts involve basic operations like addition, subtraction, multiplication, and division. Multiplication, one of these core operations, plays a foundational role in arithmetic.
Multiplicative operations rely on understanding several important concepts:
  • **Multiplication as Repeated Addition**: At its core, multiplication is simply repeated addition. For example, \(3 \times 4\) is adding 3 four times, resulting in 12.
  • **Factors and Products**: In a multiplication expression such as \(a \times b\), \(a\) and \(b\) are factors, and the outcome is the product.
  • **Use of Properties**: Applying properties like associative and distributive can simplify complex arithmetic problems. For instance, recognizing when to group factors efficiently using the associative property can speed up calculations.
Mastering these concepts ensures a thorough understanding of arithmetic and lays the groundwork for more advanced mathematics.

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Most popular questions from this chapter

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=1,000,000, r=0.1\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=2, r=-1\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

What is the common ratio in a geometric sequence?

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0004,-0.004,0.04,-0.4, \ldots\)
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