Problem 11 - Algebra 1

Question

Find the product. $$-5 \cdot 2 \cdot(-7)$$

Step-by-Step Solution

Answer

The product of -5, 2, and -7 is 70

Step 1: Multiply First Two Numbers

The first step is to multiply the first two numbers, -5 and 2. Using the multiplication rule for a positive and a negative number (the product of a positive number and a negative number is a negative number), the result is \( -5 \cdot 2 = -10 \)

Step 2: Multiply the Result by the Third Number

The next step is to multiply the result from Step 1 by the third number, -7. The multiplication rule for two negative numbers is that their product is a positive number, therefore the result is \( -10 \cdot -7 =70 \)

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are commonly used to represent things like debts or temperatures below freezing. One of the key concepts with negative numbers is that:

• Adding a negative number is like subtracting a positive number.
• Subtracting a negative number is equivalent to adding a positive number.
• Multiplying or dividing a negative number by a positive number gives a negative result.
• Multiplying or dividing two negative numbers results in a positive number.

The rules of operations with negative numbers are essential for understanding arithmetic involving them. Recognizing when and how to apply these rules is pivotal, as they help maintain the balance and order of operations in arithmetic sequences.

Key Multiplication Rules

Multiplication has predictable rules that greatly aid in arithmetic. Understanding these rules can make solving problems involving multiplication much clearer. Here are the basic multiplication rules:

• The product of a positive number and another positive number is positive.
• The product of a positive number and a negative number is negative.
• The product of two negative numbers is positive.

Let's apply these to our example. For \(-5 \cdot 2\), the result is \(-10\) because a negative times a positive is a negative. When we then take \(-10\) and multiply by \(-7\), the result becomes \(70\) as both numbers are negative, resulting in a positive product. Remembering these rules simplifies problems and avoids mistakes.

Order of Operations in Multiplication

The order of operations provides guidance on which calculations to perform first when solving an expression. For multiplication, knowing when to multiply can change the final result of a problem. With integers, the process generally follows the basic order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In our specific problem, we follow these steps:

• First, multiply from left to right. Starting with \(-5 \cdot 2\) because it's the first operation, giving us \(-10\).
• Second, take the result \(-10\) and multiply by the next number, \(-7\), yielding \(70\).

The systematic approach ensures accurate results, irrespective of the complexity of the expression. Thus, adhering properly to the order of operations will help avoid errors and achieve the correct solution.