Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-2)(-7)(-1)(3)$$

Step-by-Step Solution

Answer

The result of the multiplication (-2)(-7)(-1)(3) is -42.

Step 1: Identify the given numbers

The numbers given are -2, -7, -1, and 3. All of these numbers need to be multiplied together.

Step 2: Apply rule of multiplication for negative numbers

The rule states, when two negative numbers are multiplied together, the result is a positive number. So, multiply -2 and -7 together first: \((-2)(-7) = 14\). Then multiply 14 with -1: \(14(-1) = -14\).

Step 3: Multiply the result with the remaining number

Now multiply -14 (which is the result from step 2) with the remaining number which is 3: \(-14(3) = -42\).

Key Concepts

Integer Multiplication

Multiplying integers, whether positive or negative, follows specific rules that help us find the result. To master integer multiplication, one should start by understanding how multiplication works with positive numbers. With positive numbers, multiplication is essentially the process of repeated addition. For example, the product of two positive numbers would look something like this: \( 4\times 3 = 4 + 4 + 4 = 12 \).When we extend this to include negative numbers, the rules adjust slightly. An even number of negative factors will result in a positive product, while an odd number of negative factors will yield a negative product. For instance:- Multiplying two negative integers, like \( (-2)\times(-3) \) results in \( 6 \) because negative times negative gives a positive.- Adding another negative to the mix, such as \( (-2)\times(-3)\times(-1) \) would result in \( -6 \) since we now have an odd number of negatives.Always remember to perform multiplication operations from left to right following the order of operations, unless parentheses suggest a different order. This systematic approach ensures accuracy and helps clarify the process for beginners or those who may struggle with multiplication concepts.

Negative Number Rules

Understanding the rules for negative numbers is essential, not just for multiplication, but for all algebraic operations. First and foremost, a negative number is the opposite of a positive number and is represented by a minus sign \( - \). Here are some foundational rules:

• A negative times a negative equals a positive (as previously explained in integer multiplication).
• A negative times a positive equals a negative.
• Dividing follows the same sign rules as multiplication; negative divided by negative is positive, and negative divided by positive is negative.
• The addition of negative numbers involves subtracting their absolute values and attaching the negative sign to the result.
• To subtract a negative number, you essentially add its positive counterpart due to the 'minus a minus' rule.

Always remember to be mindful of these rules while performing operations with negative numbers to avoid common mistakes. Practice with a variety of examples, both simple and complex, to reinforce these concepts and improve your expertise in dealing with negative numbers.

Algebraic Operations

Algebraic operations are the foundation of algebra and include addition, subtraction, multiplication, and division applied to algebraic expressions and equations. Each operation follows particular properties and rules to maintain mathematical integrity. Some key properties include:

• Commutative property: Switching the order of addends or factors does not change the sum or product (applicable to addition and multiplication).
• Associative property: The grouping of addends or factors can be changed without affecting the sum or product (also applicable to addition and multiplication).
• Distributive property: A multiplied value can be distributed over terms that are added together inside parentheses, for example, \( a(b + c) = ab + ac \).

In the case of the exercise provided, using these properties helps to unravel and solve complex problems by breaking them down into simpler parts; this is especially useful when dealing with integers and their respective rules. Regular practice of these algebraic operations while adhering to the established rules is crucial for students to enhance their problem-solving skills and algebraic understanding.