Problem 40 - Elementary and Intermediate Algebra

Question

Multiply. $$ 9 \cdot(-2) \cdot(-6) \cdot 7 $$

Step-by-Step Solution

Answer

756

Step 1 - Multiply the first two numbers

Start by multiplying the first two numbers: \(9\) and \(-2\). \[ 9 \times (-2) = -18 \]

Step 2 - Multiply the result by the next number

Multiply the result from Step 1 by the next number \(-6\). \[ -18 \times (-6) = 108 \] Remember that multiplying two negative numbers results in a positive number.

Step 3 - Multiply by the final number

Now multiply the result from Step 2 by the last number \(7\). \[ 108 \times 7 = 756 \]

Key Concepts

Negative Numbers

In mathematics, negative numbers are numbers less than zero. They are written with a minus sign (-) in front of them. For example, -2 and -6 are negative numbers. These numbers can represent a decrease, a loss, or something going in the opposite direction. When dealing with negative numbers, remember:
- A negative number multiplied by another negative number results in a positive number.
- A negative number multiplied by a positive number results in a negative number.
This can be helpful when working on problems like the one provided, where you see negative numbers such as (-2) and -6).

Order of Operations

The order of operations is essential in mathematics to ensure expressions are solved correctly. This set of rules tells us the sequence in which operations should be performed to get the right answer. The common acronym to remember is PEMDAS:

• P: Parentheses first
• E: Exponents (such as squares)
• M and D: Multiplication and Division (from left to right)
• A and S: Addition and Subtraction (from left to right)

In the provided exercise, we're focusing on the multiplication aspect. Start with the first pair of numbers, continue to the next, and so on. In our case, you would:

• Multiply 9 and -2 to get -18
• Then multiply -18 by -6 to get 108, because a negative times a negative is a positive.
• Finally, multiply 108 by 7 to get 756.

Following the order of operations ensures accuracy and avoids errors.

Positive and Negative Multiplication

Understanding how to handle positive and negative numbers during multiplication is crucial. Here are the simple rules:

• Positive times Positive equals Positive: \[ 9 \times 7 = 63 \]
• Negative times Negative equals Positive: \[ -2 \times -6 = 12 \]
• Positive times Negative equals Negative: \[ 9 \times -2 = -18 \]
• Negative times Positive equals Negative: \[ -18 \times 7 = -126 \]

With these rules in mind, you can solve the given problem step-by-step as follows:

• Multiply 9 and -2 to get -18.
• Then multiply -18 and -6 to get 108 (two negatives give a positive).
• Finally, multiply 108 and 7 to get 756.

These steps ensure you correctly solve problems involving both positive and negative numbers during multiplication.