Problem 66 - Prealgebra

Question

The problems below review some of the properties of addition and multiplication we covered in Chapter 1. Rewrite each expression using the commutative property of addition or multiplication. $$4 y$$

Step-by-Step Solution

Answer

\(4y\) can be rewritten as \(y \cdot 4\).

Step 1: Understanding the Commutative Property

The commutative property states that the order of adding or multiplying numbers does not change their sum or product. For addition, it is: \(a + b = b + a\). For multiplication, it is: \(a \times b = b \times a\).

Step 2: Apply the Commutative Property to the Given Expression

We have the expression \(4y\), which involves multiplication. According to the commutative property of multiplication, \(4y\) can be rewritten as \(y \times 4\).

Step 3: Rewriting the Expression

Using the commutative property, rewrite \(4y\) as \(y \times 4\) or simply \(y \cdot 4\).

Key Concepts

Properties of Addition

In mathematics, the properties of addition make it easier for us to work with numbers efficiently and accurately, especially in complex expressions. One of the key properties is the commutative property, which states that the order of numbers in an addition operation does not affect the sum. For example, if you have two numbers, say 5 and 3, adding them either as \(5 + 3\) or \(3 + 5\) will give you the same result, which is 8.

This property is extremely useful when solving expressions because it gives you the flexibility to rearrange numbers for easier calculation. Other important properties of addition include the associative property (changing groupings doesn't affect the sum) and the identity property of addition (adding zero leaves the number unchanged).

Understanding these properties allows students to simplify expressions swiftly and identify different ways to approach a problem with confidence. It’s like having a toolbox that offers multiple solutions!

Properties of Multiplication

Just like addition, multiplication comes with its own set of useful properties. The commutative property of multiplication suggests that the order of factors does not change the product. This means that for any numbers \(a\) and \(b\), \(a \times b = b \times a\).

For example, if you multiply 7 by 2, you will get the same answer when you multiply 2 by 7, which is 14. This property is especially handy when dealing with larger numbers or algebraic expressions, as it allows flexibility in calculations.

The multiplication properties don’t stop there. We also have the associative property (regrouping numbers doesn’t change the product) and the distributive property (multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then adding the results). Each property gives us a stronger command over mathematical expressions and problem-solving.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operation symbols that represent a value. For instance, expressions like \(4 + 5\) or \(3x + 2y\) demonstrate how numbers and symbols work together.

When we use expressions, we aim to simplify or solve them to find a particular value or to express an idea concisely. Given expressions might involve different operations, and utilizing properties such as commutative, associative, and distributive properties helps us manage and simplify them.

In the exercise we examined, the expression \(4y\) uses the commutative property of multiplication to be rewritten as \(y \times 4\) or \(y \cdot 4\). This flexibility makes mathematical expressions manageable, and understanding the rules behind them equips students to manipulate and solve expressions efficiently in diverse mathematical tasks.