Problem 9
Question
State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$12 y x=12 x y$$
Step-by-Step Solution
Verified Answer
The commutative property of multiplication.
1Step 1: Identify the Operation
In the expression \(12 y x = 12 x y\), the operation involved is multiplication.
2Step 2: Recall Properties of Multiplication
There are several properties of multiplication: the commutative property, the associative property, and the identity property.
3Step 3: Apply the Relevant Property
The commutative property of multiplication states that changing the order of the factors does not change the product. Since \(12 y x = 12 x y\) rearranges the factors \(y\) and \(x\), this is justified by the commutative property of multiplication.
Key Concepts
Properties of MultiplicationCommutative PropertyAssociative PropertyEducational Mathematics
Properties of Multiplication
Understanding the properties of multiplication is essential in mathematics. These properties help simplify and solve multiplication problems, making math more manageable. Here are the main properties you need to know:
- Commutative Property: This property tells us that the order of numbers does not affect the product. Essentially, it means you can multiply numbers in any order, and the result will be the same. For example, \( a \times b = b \times a \).
- Associative Property: This property indicates that when three or more numbers are multiplied, the way they are grouped does not affect the product. For instance, \((a \times b) \times c = a \times (b \times c)\).
- Identity Property: This property states that any number multiplied by 1 remains unchanged, such as \( a \times 1 = a \).
Commutative Property
The commutative property of multiplication is a crucial concept. It states that swapping the order of numbers in a multiplication equation does not change the product. This property is particularly helpful when solving algebraic expressions or rearranging terms for simplification.
Think of it like rearranging items in a bag. No matter which item you put in first, your bag still contains the same items in total. Mathematically, this means for any numbers \(a\) and \(b\), \(a \times b = b \times a\).
An example of the commutative property in action is the expression \(12y \times x = 12x \times y\). Here, rearranging \(y\) and \(x\) does not affect the product. This property simplifies calculations and helps verify that rearranging does not lead to mistakes.
Think of it like rearranging items in a bag. No matter which item you put in first, your bag still contains the same items in total. Mathematically, this means for any numbers \(a\) and \(b\), \(a \times b = b \times a\).
An example of the commutative property in action is the expression \(12y \times x = 12x \times y\). Here, rearranging \(y\) and \(x\) does not affect the product. This property simplifies calculations and helps verify that rearranging does not lead to mistakes.
Associative Property
The associative property of multiplication is another important mathematical principle. It states that when you multiply three or more numbers, the way you group them does not change the product. This means that you can change the grouping of numbers, and it won't affect the outcome.
To get a clearer picture, consider the numbers \(a, b, \text{ and } c\). The property tells us that \((a \times b) \times c = a \times (b \times c)\). The result remains constant irrespective of how the numbers are grouped.
This property is incredibly useful when dealing with complex multiplication problems or algebraic expressions. It allows flexibility in calculation, enabling you to perform operations in the most convenient and efficient order.
To get a clearer picture, consider the numbers \(a, b, \text{ and } c\). The property tells us that \((a \times b) \times c = a \times (b \times c)\). The result remains constant irrespective of how the numbers are grouped.
This property is incredibly useful when dealing with complex multiplication problems or algebraic expressions. It allows flexibility in calculation, enabling you to perform operations in the most convenient and efficient order.
Educational Mathematics
Educational mathematics plays a pivotal role in developing logical and analytical thinking. Core concepts like the properties of multiplication lay a foundation for more advanced topics. When students understand these properties, they gain the ability to simplify and solve complex problems with ease.
Learning these foundational concepts also promotes better reasoning and problem-solving skills. Mathematics education focuses on using properties like the commutative and associative properties to foster a deeper understanding of algebra and arithmetic.
By mastering these properties early on, students can approach more challenging mathematical concepts with confidence, ultimately leading to success in their educational journey. Remember, practice is key to reinforcing these concepts and enhancing your math skills.
Learning these foundational concepts also promotes better reasoning and problem-solving skills. Mathematics education focuses on using properties like the commutative and associative properties to foster a deeper understanding of algebra and arithmetic.
By mastering these properties early on, students can approach more challenging mathematical concepts with confidence, ultimately leading to success in their educational journey. Remember, practice is key to reinforcing these concepts and enhancing your math skills.
Other exercises in this chapter
Problem 8
Identify each statement as true or false. Zero is a rational number.
View solution Problem 9
Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$-3 a^{2}+7 b^{2}+9 a^{2}-2 b^{2}$$
View solution Problem 9
Perform the following operations with real numbers. $$(-9)(-12)$$
View solution Problem 9
Identify each statement as true or false. All whole numbers are integers.
View solution