Problem 141 - Algebra A Combined Function

Question

Write each as an algebraic expression. Then simplify the expression. -1 added to the product of -8 and -5

Step-by-Step Solution

Answer

The simplified algebraic expression is 39.

Step 1: Translate Words to Symbols

First, we need to translate the given words into an algebraic expression. "Product of -8 and -5" means we multiply these two numbers: \(-8 \times -5\). Then "-1 added to" means we will add -1 to the result of that product.

Step 2: Calculate the Product

Now we calculate the product of -8 and -5. Since both numbers are negative, their product is positive:\[-8 \times -5 = 40\]

Step 3: Add -1 to the Product

Next, add -1 to the product we found previously:\[40 + (-1) = 40 - 1\]

Step 4: Simplify the Expression

Finally, perform the subtraction to simplify the expression:\[40 - 1 = 39\]

Key Concepts

Product of Numbers

Understanding the concept of "product" is fundamental in algebra. When we talk about the "product of numbers," we mean the result you get when you multiply numbers together. In this instance, we needed to find the product of -8 and -5. Multiplying these numbers involved the rule for multiplying negative numbers, which is critical to remember. When both numbers are negative, as -8 and -5 are, their product becomes positive. This is because the multiplication of two negative numbers equals a positive, following the rule of signs:

• Negative × Negative = Positive
• Negative × Positive = Negative
• Positive × Negative = Negative
• Positive × Positive = Positive

So, \[-8 \times -5 = 40\].Remember, matching the signs when multiplying can make determining the sign of the product much easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining their original value. In algebra, we always aim to express our answers in the most straightforward way possible. For this exercise, simplifying the expression "product of -8 and -5, then add -1" involves a few steps.
First, calculate the intermediate product, which we found to be 40. Then \(40 + (-1)\) signifies adding -1 to the product. When simplifying, notice that adding a negative number is similar to subtracting its positive: \(40 + (-1) = 40 - 1\). This gives you the simplified expression of 39. Simplification is essential because it makes expressions easier to work with and clarity is critical in complex problems.

Negative Numbers

Negative numbers are a vital part of mathematics as they expand the number system beyond zero to include values less than zero. Understanding how they interact with one another is crucial. In this exercise, both -8 and -5 were negative. It's important to recall that when two negative numbers are multiplied, their product is positive. Conversely, adding negative numbers means that we typically move further left on the number line, which effectively reduces the total value.
For example, instead of adding negatively, think of it as removing from the positive you had entirely:

• If you have 40, adding -1 takes you to 39, like a simple subtraction.

Negative numbers can be tricky initially, but with practice, they become an intuitive part of solving algebraic problems.

Adding Integers

Adding integers is another fundamental topic in algebra. Integers include all whole numbers and their negative counterparts. When adding integers, especially those with different signs, we apply several straightforward rules. Consider this:

• Adding a negative number of the same value cancels part of a positive integer (e.g., \(5 + (-5) = 0\)).
• To add a negative integer is to subtract its absolute value from the positive integer (e.g., adding -1 is like subtracting 1).

In our original problem, \(40 + (-1)\) effectively becomes \(40 - 1 = 39\). Understanding these principles helps in rapidly addressing problems where integers require addition in different contexts.