Problem 32 - Elementary Algebra

Question

Simplify. $$ (-7) 3 $$

Step-by-Step Solution

Answer

-21

Step 1: Understanding the Problem

The expression \[ (-7) imes 3 \]simply requires multiplying two numbers: \(-7\) and \(+3\).

Step 2: Performing the Multiplication

Multiply \(-7\) by \(3\):Negative multiplied by positive results in a negative number.\[ (-7) imes 3 = -21 \]

Step 3: Conclusion

Therefore, the simplified form of the expression is \(-21\).

Key Concepts

Negative Numbers

Negative numbers represent quantities less than zero. They are often used in mathematics to express loss, debt, or direction. Negative numbers are written with a minus sign (e.g., -7). When dealing with calculations involving negative numbers, it's essential to understand how they interact with positive numbers and each other.

• If you multiply a negative number by a positive number, the result is always negative. For example, \((-7) \times 3 = -21\).
• When you multiply two negative numbers, the result is positive. For example, \((-4) \times (-5) = 20\).
• Addition and subtraction involving negative numbers also have specific rules. Subtracting a negative is like adding a positive, e.g., \(-5 - (-3) = -5 + 3 = -2\).

Understanding negative numbers helps simplify calculations and solve various mathematical problems confidently.

Order of Operations

In mathematics, the order of operations is a fundamental principle that guides how expressions should be simplified or evaluated. This principle ensures consistency, so everyone gets the same result when solving an expression. The standard order of operations can be remembered by the acronym PEMDAS:

• Parentheses: First, simplify expressions inside parentheses.
• Exponents: Evaluate powers and square roots after parentheses.
• Multiplication and Division: Proceed from left to right.
• Addition and Subtraction: Also from left to right.

This order might not affect simple arithmetic operations, like multiplying a single integer by negative numbers. However, in complex expressions, it is crucial. Even simple tasks like multiplying \((-7) \times 3\) implicitly follow these rules, as multiplication comes before any addition or subtraction in PEMDAS.

Integer Arithmetic

Integer arithmetic deals with whole numbers, which include positive numbers, negative numbers, and zero. It is crucial in basic mathematics and forms the foundation for more complex mathematical theories.When performing integer arithmetic, addition, subtraction, multiplication, and division are common operations:

• Addition: Adding integers with different signs involves finding the difference and giving the result the sign of the larger absolute value. For example, \(-4 + 2 = -2\).
• Subtraction: Subtracting integers can be seen as adding the opposite. For instance, \(-7 - 3 = -10\).
• Multiplication: As noted, negative times positive is negative, while positive times negative is also negative.
• Division: Division of integers follows the same sign rules as multiplication. A negative divided by a positive gives a negative result, e.g., \(-9 / 3 = -3\).

Integer arithmetic's clear rules make calculations easier and help in solving real-world problems, from financial calculations to scientific computing.