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Chapter 1: Problem 76

Rewrite each of the following using the commutative property of multiplication. 4(3)

Short Answer

Expert verified
The expression 4(3) can be rewritten as 3(4) using the commutative property of multiplication.

Step by step solution

01

Understanding the Commutative Property

The commutative property of multiplication states that changing the order of the factors does not change the product. Mathematically, for any numbers \(a\) and \(b\), \(a \times b = b \times a\).
02

Applying the Commutative Property

In the given expression, 4(3), we have two factors: 4 and 3. According to the commutative property, we can rewrite it as 3(4).
03

Rewrite the Expression

Using the commutative property, rewrite the expression 4(3) as 3(4).
04

Verification

To verify, calculate both expressions: 4 times 3 equals 12, and 3 times 4 also equals 12. This confirms that the expressions are equivalent due to the commutative property of multiplication.

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Key Concepts

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

Factors Multiplication
When we talk about multiplying factors, we're referring to the building blocks of an expression that involve numbers being multiplied. In the expression \(4(3)\), the numbers 4 and 3 are known as factors. Multiplication is simply the process where these factors are combined to find a total or a product.

Whenever you see numbers like 4 and 3 together with a multiplication sign or in parentheses as 4(3), you're tasked with finding how many times one number is contained in the other. In school, you might have learned multiplication tables, which help in swiftly determining products like 4 times 3, which equals 12.
  • You can think of multiplication as repeated addition, for example, 4 times 3 is adding 4 three times (4 + 4 + 4).
  • Factors are flexible in order, meaning the product remains the same as you'll discover with equivalent expressions.
Equivalent Expressions
Equivalent expressions are expressions that look different but represent the same value. In mathematics, discovering equivalent expressions is essential because it helps in simplifying problems and finding solutions in different forms.

When you use the commutative property of multiplication on the expression \(4(3)\), you swap the factors to get \(3(4)\). Both expressions are equivalent because they equal 12.
  • To check if expressions are equivalent, compute the value of each to ensure they match. This is what we did with \(4 \times 3\) and \(3 \times 4\).
  • Equivalent expressions play a key role in algebra, allowing you to manipulate equations for easier solving.
Mathematical Properties
Mathematical properties are rules that apply to numbers and operations, helping to simplify and solve problems. The most common properties include commutative, associative, and distributive properties. The focus here is on the commutative property.

The commutative property of multiplication tells us that the order in which numbers are multiplied does not change their product. That is, \(a \times b = b \times a\) for any numbers \(a\) and \(b\).
  • This property is not only useful in theoretical mathematics but also simplifies real-world calculations.
  • By changing the order of multiplication, you might find easier mental calculations or algebraic simplifications.

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Most popular questions from this chapter

Divide. You shouldn't have any wrong answers because you can always check your results with multiplication. $$4 5 \div { 1 3 5 , 9 0 0 }$$

Rewrite each of the following using the associative property of addition. $$2+(8+2)$$

Add each of the following. (All problems involve carrying in at least one column. $$\begin{array}{r} 444 \\ 595 \\ \hline \end{array}$$

Rewrite each of the following using the associative property of addition. $$(2+3)+8$$

Add each of the following. (All problems involve carrying in at least one column. $$\begin{array}{r} 439 \\ 270 \\ \hline \end{array}$$
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