Problem 99 - Algebra 1

Question

Evaluate the expression. $$-8 \cdot(-9)-80$$

Step-by-Step Solution

Answer

The result of the expression \(-8 \cdot(-9)-80\) is \(-8\).

Step 1: Apply the rule of sign

The expression is \(-8 \cdot(-9)-80\). In this step, the multiplication operation between \(-8\) and \(-9\) is performed first. The multiplication of two negative numbers gives a positive result. Therefore, \(-8 \cdot -9\) equals \(72\). The resulting expression is \(72 - 80\).

Step 2: Perform the subtraction

Now, subtract 80 from 72. \(72 - 80\) equals \(-8\).

Key Concepts

Rules of Signs

Understanding the rules of signs is critical when evaluating expressions involving multiplication and addition or subtraction of negative numbers. Essentially, these rules tell us how to handle the signs when combining numbers.

Here's a quick rundown:

• A positive number multiplied by a positive number gives a positive result.
• A negative number multiplied by a negative number also gives a positive result.
• A positive number multiplied by a negative number gives a negative result.
• A negative number multiplied by a positive number gives a negative result.

When we add or subtract numbers:

• Subtracting a positive number is the same as adding its negative counterpart.
• Adding two negative numbers together gets more negative.
• Subtracting a negative number is like adding its positive counterpart.

In the exercise \( -8 \cdot(-9)-80 \), these rules help us predict that the product of the two negative numbers, \( -8 \cdot -9 \) is positive.

Multiplication of Negative Numbers

Multiplication of negative numbers can be counterintuitive until you remember one simple rule: multiplying two negative numbers results in a positive number. This may seem odd at first, but it makes sense when you consider that a negative number is essentially the opposite of a positive number.

So when you multiply \( -8 \cdot -9 \), think of it as taking away eight sets of negative nine. Removing a negative can be thought of as adding a positive. So you're adding positive nines, eight times, which gives you \( +72 \). Understanding this concept is crucial for simplifying expressions and solving more complex algebra problems.

Arithmetic Operations

Arithmetic operations form the foundation of most mathematical expressions and include addition, subtraction, multiplication, and division. When evaluating an expression, it's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).

In the case of our exercise, multiplication comes before subtraction as per these guidelines. Therefore, you first calculate the product of the negative numbers, and then perform the subtraction. This ensures that every step of the operation is executed correctly and maintains the integrity of the mathematical expression.

Simplifying Expressions

Simplifying expressions is all about reducing a complex equation down to its simplest form. This often involves combining like terms, applying the distributive property, and following the rules of signs and arithmetic operations. The goal is to make the expression as straightforward as possible for easy interpretation and solution.

Through this concept, we can refine our exercise expression step by step, starting with multiplication and then moving to subtraction. After performing \( -8 \cdot -9 \), we simplify \( 72 - 80 \) to get \( -8 \) as the final answer. With each step, the expression becomes easier to read and understand, proving the value of simplifying in the process of solving mathematical problems.