Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$-2(-3)(-4)(-1)$$

Step-by-Step Solution

Answer

The solution to the multiplication -2(-3)(-4)(-1) is 24.

Step 1: Multiplication of the First Two Numbers

We start the process by multiplying the first two numbers together, which are -2 and -3. Applying the rule that the multiplication of two negative numbers is a positive number, we get -2 multiplied by -3 equals 6.

Step 2: Multiplication of the Result with the Third Number

Next, we multiply the result of the first step, which is 6, with the third number, which is -4. Multiplying a positive number by a negative number yields a negative number, this yields -24.

Step 3: Multiplication of the Result with the Fourth Number

Finally, we multiply the result of the second step, which is -24, with the fourth number, -1. Applying the rule that negative multiplied by a negative is positive, this yields a final result of 24.

Key Concepts

Negative Numbers in Multiplication

Negative numbers play a crucial role in multiplication. They can change the outcome's sign depending on the number of negative numbers you multiply. Let's explore how that works:

Recall that a negative number can be thought of as an opposite or direction on the number line. For example, -3 is three units away from zero, but in the opposite direction to 3.

When multiplying negative numbers, observe these key rules:

• Multiplying two negative numbers results in a positive number, because reversing direction twice brings us back to a positive.
• Multiplying a negative number by a positive number yields a negative result. This means we continue to move in the negative direction.

Using these principles helps us anticipate the result of arithmetic involving negatives, ensuring that our calculations are on track.

Basic Multiplication Rules

Understanding multiplication rules is essential to solve arithmetic problems, particularly involving multiple numbers. Multiplication is essentially repeated addition, but with its specific rules.

When you're multiplying several numbers together, consider these multiplication rules:

• If you multiply two positives, the product is always positive.
• The product of two negative numbers is positive.
• Multiplying a positive and a negative number results in a negative.
• The order doesn't matter in multiplication, due to its commutative property \(a \times b = b \times a\).

These rules help simplify expressions and guide you in calculating the right product of numbers, no matter their sign.

Combining Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental in mathematics. Integrating these operations, especially multiplication, can be tricky when negative numbers are involved.

When performing arithmetic operations, it's important to follow an order of operations (PEMDAS/BODMAS), which stands for:

• Parentheses or Brackets
• Exponents or Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When you're multiplying several numbers, focusing on two at a time while following the sign rules ensures accuracy. Always keep an eye on the number signs, as this affects the intermediate results and the final outcome. Proper handling of the arithmetic operations allows you to streamline solving complex problems with confidence.