Problem 16 - Beginning and Intermediate Algebra

Question

Find each product. \(-9(-5)\)

Step-by-Step Solution

Answer

45

Step 1: Understand the Problem

We need to find the product of two negative numbers: \(-9\) and \(-5\).

Step 2: Recall the Rule for Multiplying Negatives

The product of two negative numbers is a positive number. Therefore, \(-a\) \times \(-b\) = \(a\) \times \(b\).

Step 3: Multiply the Absolute Values

Multiply the absolute values of the given numbers: \(9\) and \(5\). So, \(9 \times 5 = 45\).

Step 4: Apply the Sign Rule

Since we are multiplying two negative numbers, their product is positive. Hence, the product of \(-9\) and \(-5\) is \(45\)

Key Concepts

Product of Integers

When it comes to multiplying integers, it is important to focus on the rules of signs and understanding the fundamental concepts. Multiplying two numbers means you're finding the total number of things in equal-sized groups.

For example, if you multiply positive integers 3 and 4, you're essentially adding the number 3 four times, which results in 12. Similarly, when multiplying negative numbers, you follow slightly different rules, but the principle of combining groups still applies. In our exercise, we need to multiply \(-9\) and \(-5\).

The product of two numbers, regardless of their signs, is the result of multiplying their absolute values — a concept we'll explain next.

Absolute Value

The absolute value of a number is its distance from 0 on the number line. It is always a positive number or zero. Absolute value is denoted by vertical bars around the number, like this: \| a \|.

For instance, \| -7 \| equals 7, and \| 9 \| equals 9.

In our exercise, we are dealing with the numbers \(-9\) and \(-5\). To find their product, we first find their absolute values:

• The absolute value of \-9\ is 9.
• The absolute value of \-5\ is 5.

Once we have these absolute values, we can multiply them together: \( 9 \times 5 = 45 \).

Multiplication Rules

To correctly find the product of two integers, understanding the multiplication rules for signs is vital. Here are the key rules to remember:

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Positive = Negative
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Negative = Positive

Remember, when you multiply two negative numbers, like in our exercise with \(-9\) and \(-5\), the negatives cancel out. This makes their product a positive number.

So, \(-9 \times -5 = 45\).

By following these multiplication rules, you can easily determine the sign of the product.