Problem 7 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-9(-9)$$

Step-by-Step Solution

Answer

The product of \(-9\) and \(-9\) is \(81\).

Step 1: Identify the numbers to be multiplied

Determine which two numbers need to be multiplied. In this case, the numbers are \(-9\) and \(-9\).

Step 2: Understand the multiplication rule for negative numbers

Recall that multiplying two negative numbers yields a positive result. This is a fundamental rule in arithmetic to avoid incorrect calculations.

Step 3: Calculate the absolute values

Compute the product of the absolute values of the numbers: \(9 \times 9 = 81\).

Step 4: Apply the sign rule

Since both numbers have negative signs, their product will be positive. So the answer is positive \(81\).

Key Concepts

Negative Number Rules

When dealing with negative numbers, it's crucial to understand some basic rules that make operations easier. Negative numbers are numbers less than zero and are typically used to represent loss, debt, or temperatures below freezing, among other things.

Here are a few important rules to remember:

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers also results in a positive product. This might be counterintuitive at first but think of it this way: two negatives make a positive. So, when you multiply -9 and -9, the negative signs cancel out, resulting in a positive 81.
• However, if you multiply a positive number by a negative number, the product is negative. The negative sign doesn't have another to cancel it out, like in the second rule.

These rules help learners avoid common mistakes when working with different types of numbers.

Absolute Values in Multiplication

Absolute value is a concept that measures the 'magnitude' of a number without considering its sign. Whenever you see an absolute value, it means you should treat the number like it is always positive.

In multiplication, especially when dealing with negative numbers, calculating with absolute values first can simplify your work. The absolute value of a number is written as: \(|x|\), and in practice: \(|-9| = 9\).

When multiplying two negative numbers like \(-9\) and \(-9\), you first multiply the absolute values: \(9 \times 9\). This step disregards the negative signs and focuses on the numbers themselves, simplifying the multiplication process to find that \(9 \times 9 = 81\). Afterward, consider the rules of negatives to determine the sign of the final answer.

Arithmetic Operations

Arithmetic operations are the basis of mathematics and include addition, subtraction, multiplication, and division. Multiplication, specifically, involves calculating the total when one number is replicated across another.

In our example, -9 multiplied by -9, we use basic multiplication of individual numbers. The process involves:

• Writing each number with their corresponding signs.
• Ignoring the signs initially and performing the multiplication on the absolute values.
• Applying the rule for negative numbers to finalize the product's sign.

Multiplication can become tricky when negative numbers are involved, but by systematically applying arithmetic operations and understanding number rules, it can be made easier. The product of -9 and -9 is positive 81 thanks to our knowledge of these operations.