Problem 10 - Algebra 1

Question

Find the product. $$-30 \cdot 8$$

Step-by-Step Solution

Answer

The product of -30 and 8 is -240.

Step 1: Identify the Numbers

Identify the numbers to be multiplied. In this case, the numbers are -30 and 8.

Step 2: Multiply the Numbers

Next, multiply the two numbers together. When multiplying -30 and 8, calculate the product as if both numbers were positive, which would be \(30 \times 8 = 240\).

Step 3: Apply Sign Rule

Since the rule dictates that the product of a negative number and a positive number is a negative number, the product of -30 and 8 is -240.

Key Concepts

Negative Number Multiplication

Understanding multiplication with negative numbers might seem tricky at first, but once you get the hang of it, it's pretty straightforward. When you multiply a negative number by a positive one, like in the example \( -30 \times 8 \), you use the same process as you would with two positive numbers but need to include one extra step to determine the sign of the answer.

For instance, let's completely ignore the minus sign for a moment and just multiply 30 and 8. If we do that, we'll get the product of 240. However, this isn't our final answer because we haven't considered the negative sign yet. To get the correct answer we need to apply the sign rules of multiplication, which we'll discuss in the next section, concluding that the final product should be negative. So, the answer to \( -30 \times 8 \) is \( -240 \).

Sign Rules in Multiplication

The rules for signs in multiplication are key to correctly solving problems involving negative numbers. Here is the crux of it: when you multiply two numbers, the result will be positive if the signs of both numbers are the same, and negative if the signs are different.

Multiplying Positives and Negatives:

When one number is negative and the other positive, as in \( -30 \times 8 \), your answer will be negative. On the other hand,

Multiplying two Negative Numbers:

If both numbers were negative, for example \( -30 \times -8 \), the negatives would cancel each other out, and your product would be positive, resulting in 240. This might seem non-intuitive, but remember, a negative multiplied by a negative equals a positive.

Basic Arithmetic

The bedrock of solving any mathematical expression, including multiplication with negative numbers, lies in mastering basic arithmetic – addition, subtraction, multiplication, and division.

In the context of multiplication, be sure to:

• Understand the multiplication table for single-digit numbers.
• Remember that any number multiplied by zero results in zero.
• Know that multiplying by one leaves the number unchanged (identical property of one).

Apprehending these basics will provide the necessary foundation for applying more complex rules, such as those involved in negative number multiplication. Mastery of basic arithmetic is the foundation upon which more advanced mathematical concepts are built.