Problem 38 - Elementary Algebra

Question

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations. $$(21 c)(0.008)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression (21c)(0.008) using the commutative property of multiplication without performing any calculations. Answer: (0.008)(21c)

Step 1: Understand the commutative property of multiplication

The commutative property of multiplication states that for any two numbers \(a\) and \(b\), the product \(ab\) is equal to the product \(ba\). In other words, you can change the order of factors in a multiplication problem without changing the result.

Step 2: Apply the commutative property to the expression

We are given the expression \((21c)(0.008)\). Let \(a = 21c\) and \(b = 0.008\). Then, using the commutative property of multiplication, we can rewrite the expression as \((0.008)(21c)\). So, the expression \((21c)(0.008)\) is equal to the expression \((0.008)(21c)\).

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation, commonly used to find the total of items grouped into equal sets. Imagine having 3 bags, each with 5 apples; by multiplying 3 (bags) by 5 (apples per bag), you find out that there are 15 apples in total.
In algebra, multiplication can involve numbers, variables, or both. For instance, multiplying numbers like 4 and 5 is straightforward, resulting in 20. Variables, such as x and y, can be multiplied to form a product like xy.
When dealing with expressions such as \( (3x)(4y) \), you multiply the numbers together \(3 \times 4 = 12\), and then attach the variables \(x \times y = xy\), resulting in the product \(12xy\).
Multiplication is flexible due to properties like the commutative property, making calculation easier in various scenarios.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can consist of numbers, variables, and arithmetic operations. They are a crucial component in algebra used to denote equations or functions elegantly.
For example, the expression \( 4x + 7 \) includes a variable x and constants 4 and 7, along with addition and multiplication. Variables are symbols like x or y that can represent various values.
Algebraic expressions can be simple, such as \( 3x \), or more complex, like \( 5x^2 + 3xy - 7 \).
These expressions allow us to create equations and inequalities to solve real-world problems. This capability makes understanding how to manipulate and use expressions essential for advancing in mathematics.
Using algebraic expressions, for example, helps in solving for unknowns and expressing relationships between different quantities.

Properties of Operations

Properties of operations are rules that govern arithmetic operations like addition and multiplication, helping simplify complex calculations. These properties are crucial in advancing from basic arithmetic to more complex algebraic functions.
Some fundamental properties include:

• Commutative Property: This property states that the order in which two numbers are added or multiplied does not affect the result. For example, for multiplication, \( ab = ba \), as demonstrated with \((21c)(0.008) = (0.008)(21c)\).
• Associative Property: This property allows regrouping numbers when adding or multiplying without changing the result. For example, \((a + b) + c = a + (b + c)\) for addition.
• Distributive Property: This property connects multiplication and addition, allowing us to multiply a number by a group of numbers added together, e.g., \(a(b + c) = ab + ac\).

Applying these properties helps in simplifying expressions and solving algebraic equations efficiently, making them essential for anyone studying mathematics.