Problem 30 - Elementary Algebra

Question

Simplify. $$ (-2) 4 $$

Step-by-Step Solution

Answer

The simplified expression is \(-8\).

Step 1: Understand the Expression

The expression given is \((-2) \times 4\). Here, \(-2\) is multiplied by \(4\). It's a basic multiplication we need to simplify.

Step 2: Multiply the Numbers

Multiply the numbers: \(-2\) and \(4\).When multiplying a negative number by a positive number, the product is negative. So, \((-2) \times 4 = -8\).

Key Concepts

Multiplication

Multiplication is one of the basic operations in elementary algebra. It involves finding the total when one number is taken a certain number of times, which is depicted by another number. For example, when you see \(2 \ imes 3\), you're looking at taking the number 2, three times. This results in 6. In this exercise, we see \((-2) \ imes 4\), which means the number -2 is taken four times. Multiplication helps simplify expressions by eliminating repeated addition.
Here are some essential multiplication rules to remember:

• Multiplying any number by 1 gives the number itself. For instance, \(7 \ imes 1 = 7\).
• Multiplying any number by 0 yields zero. E.g., \(5 \ imes 0 = 0\).
• When multiplying two numbers, the order doesn't matter. \(3 \ imes 4 = 4 \ imes 3\).

Understanding these rules helps in simplifying and correctly executing multiplication in algebraic expressions. \((-2) \ imes 4\) is a perfect example of a straightforward multiplication that simplifies directly by applying these basic principles.

Negative Numbers

Negative numbers can initially feel tricky, but understanding them is crucial. A negative number is any number less than zero and is represented by a minus sign (-). They are used to express values that decrease, debt, or below zero temperatures, among other things. In algebra, they follow specific rules, especially when used in multiplication.
When multiplying by a negative number, the result depends on the sign of the other number involved. Here's how it works:

• Multiplying a negative number by a positive number results in a negative.For instance, \(-2 \ imes 4 = -8\).
• Multiplying two negative numbers yields a positive result. E.g., \(-3 \ imes -2 = 6\).

These rules come from the definition of multiplication as an extension of addition. Knowing how to handle negative numbers means you can simplify expressions like \((-2) \ imes 4\) effectively and confidently.

Simplification

Simplification in algebra means reducing an expression to its simplest form, making it easier to understand and work with. It involves basic algebraic operations like addition, subtraction, multiplication, and division while applying arithmetic rules.
When simplifying, follow these steps:

• Identify the numbers or variables involved in the operation.
• Apply the arithmetic operation as per the given expression.
• Consider the rules of the operation, such as those for negative numbers.

In this exercise, we simplify \((-2) \ imes 4\) by computing the result as \(-8\). This is the simplest form of the expression and is as reduced as possible. Simplification is essential in algebra as it keeps processes streamlined and requires understanding and applying the correct operation rules.