Problem 18 - Algebra 1: Concepts and Skills

Question

Find the product. \(5(-5)\)

Step-by-Step Solution

Answer

The product of \(5\) and \(-5\) is \(-25\).

Step 1: Multiply the numbers

Now, multiply the numbers disregarding the sign: \(5*5 = 25\)

Step 2: Consider the signs

Here, one of the numbers is negative, which will make the result of multiplication negative. Therefore, we get \(-25\) as a result.

Key Concepts

Understanding the Product of Integers

When we speak about the product of integers, we're talking about the result of multiplying two whole numbers. Integers can be positive, negative, or zero. The product is the total we get when one integer value is multiplied by another. It involves performing basic multiplication but with special attention to the numbers' signs.

For example, if we need to find the product of integers like 5 and -5, it's the result of multiplying these numbers together. In our original exercise, the solution required multiplying these two numbers together to find their product. Understanding how to handle both the number and its sign is key to deriving the correct result.

Explaining Sign Rules in Multiplication

When performing multiplication involving integers, sign rules are crucial for determining the correct sign of the product. Here are the essential rules you need to remember:

• When two positive integers are multiplied, the product is positive. For example, 3 multiplied by 2 gives 6.
• When two negative integers are multiplied, the product is also positive. For example, (-3) multiplied by (-2) yields 6.
• When a positive integer is multiplied by a negative integer, the product is negative. For instance, 5 multiplied by (-3) results in -15.

While these rules might seem a bit tricky at first, they're consistent. In practice, starting with knowing the multiplication of positive numbers, then applying the sign rule ensures you achieve accuracy in your calculations. This was clearly demonstrated in the original exercise, where multiplying -5 by 5 resulted in -25.

Grasping Integer Arithmetic

Integer arithmetic involves the operations you do with whole numbers, including addition, subtraction, multiplication, and division. While arithmetic with positive integers might seem straightforward, introducing negative integers demands more attention due to the sign rules.

When tackling problems in integer arithmetic, it's vital to stay organized:

• Begin with the basic operation, whether it's addition, subtraction, multiplication, or division.
• Pay close attention to the signs and apply the appropriate rules.
• Write down each step if it helps to avoid mistakes, especially when handling complex problems.

Integer arithmetic is foundational for more advanced mathematics. As with any skill, practice helps build confidence and competence. Always remember to respect the rules regarding the signs, as these can significantly impact your results – just like in the multiplication of -5 and 5 in our exercise.