Problem 25 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. 4\((2 m)\)

Step-by-Step Solution

Answer

The expression without parentheses is \(8m\).

Step 1: Distribute the Constant

To eliminate the parentheses in the expression, we need to use the distributive property of multiplication over addition. The expression given is \(4(2m)\). Since there's no addition or subtraction inside the parentheses, we'll directly multiply the constant outside (4) with each term inside the parentheses (2m).

Step 2: Multiply the Constants

Multiply 4 by 2, which results in 8. The expression now becomes \(8m\) after applying the distributive property.

Key Concepts

Real Numbers

Real numbers are the numbers that we use in everyday life. They include all the numbers that can be found on the number line.
Real numbers can be

• Positive numbers (like 2 or 3.5)
• Negative numbers (like -1 or -0.5)
• Whole numbers or integers (like 0, 1, 2)
• Fractions and decimals (like 1/2 or 0.75)

The concept of real numbers is crucial in algebra because they form the basis of all algebraic operations. In the given exercise, the number 4 and the coefficient 2 in the parenthesis are both real numbers. When using the distributive property, we multiply these real numbers together to simplify the expression without parentheses.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the expression 4(2m), we see:

• Numbers: such as 4 and 2
• A variable: 'm'
• A multiplication operation: implied between 4 and 2m

Algebraic expressions do not have an equality sign, meaning they are not equations but statements that can represent various things. The goal in algebra is often to simplify these expressions. By applying rules such as the distributive property, we can "break them down" into simpler forms, like transforming 4(2m) into 8m. This process helps make calculations easier later on.

Multiplication in Algebra

Multiplication is one of the fundamental arithmetic operations. In algebra, multiplication often appears in expressions involving variables. When you multiply numbers by a variable, you distribute each term independently.
For the expression 4(2m), multiplication involves:

• Multiplying the numerical coefficients (4 and 2) together, which gives you 8.
• Keeping the variable 'm' as it is, since it is multiplied by the result of 4 × 2.

The distributive property helps simplify such multiplication tasks by allowing you to distribute the number outside the parentheses through to the terms within. Consequently, 4(2m) becomes 8m. This technique ensures clarity and works universally, regardless of the complexity of the expressions.