Problem 41 - Elementary Algebra

Question

Simplify. $$ -(12) 3 $$

Step-by-Step Solution

Answer

The simplified form of \(-(12)3\) is \(-36\).

Step 1: Understand the Expression

The expression given is \(-(12)3\). It involves multiplying \(-(12)\) by \(3\). We need to simplify this by performing the multiplication.

Step 2: Simplify the Multiplication

Multiply \(3\) by \(-12\). The rule for multiplying integers is that a negative times a positive results in a negative number. Therefore the product is \(3 \times (-12) = -36\).

Step 3: Interpret the Result

After multiplying, we find that the simplified form of the expression \(-(12)3\) is \(-36\).

Key Concepts

Negative Numbers

Negative numbers are values less than zero, shown with a minus sign (−) in front of them. Understanding how negative numbers operate is crucial, especially when performing mathematical operations like addition and multiplication.

Negative numbers are the opposite of positive numbers. For example,

• If you have 5 and you owe 5, you are at 0. The owing part can be thought of as -5.
• When moving on a number line, moving to the left represents negative numbers.

In operations like multiplication, these numbers follow specific rules that determine the sign of the result.

Multiplying Integers

Multiplying integers is a key math skill, especially important for simplifying expressions with negatives. The rules are simple and easy to remember:

• A positive number multiplied by a positive number equals a positive number (e.g., 2 × 3 = 6).
• A negative number multiplied by a negative number also equals a positive number (e.g., ext{-}4 imes -2 = 8 ext{). This might seem counterintuitive, but think of it as the negatives canceling each other out.
• A positive number multiplied by a negative number, or a negative number multiplied by a positive number, results in a negative number (e.g., 3 × ext{-}2 = ext{-}6).

Understanding these rules helps you quickly determine the outcome of multiplying any two integers, whether they be positive or negative.

Simplification of Expressions

Simplification of expressions involves reducing them to their simplest form. It requires knowing the rules of arithmetic operations, especially when they include negative integers.

In the example given, ext{-}(12)3 ext{), the simplification process involves:

• Identifying the operation: multiplication.
• Recognizing the sign of each integer.
• Applying the multiplication rule for integers: a negative number (−12) times a positive number (3) results in a negative product (−36).

The goal of simplification is to make the expression as straightforward as possible, making it easier to understand and use in further calculations. With practice, simplifying expressions becomes a natural part of working with math.