Problem 31 - Introductory Algebra for College Students

Question

perform the indicated multiplication. $$5(-3)(-1)(2)(3)$$

Step-by-Step Solution

Answer

The result of the indicated multiplication is 90.

Step 1 Product of Negative Numbers

Start by calculating the product of two negative numbers, which is -3 and -1. This gives: \( -3 \times -1 = 3 \)

Step 2 Continuation of Multiplication

Now multiply the result from step 1 by the remaining integers: \( 3 \times 5 \times 2 \times 3 = 90 \)

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are often used to indicate a debt or a deficit. Handling negative numbers can sometimes be tricky, especially when performing arithmetic operations. There are key rules when multiplying negative numbers that you should remember:

• When you multiply two negative numbers, the result is a positive number. This is because a negative times a negative equals a positive.
• If you multiply a positive number by a negative number, the result will be negative.
• Multiplying several numbers where there is an odd count of negatives will result in a negative product. An even count results in a positive product.

So, in our original exercise, multiplying (-3) and (-1) results in a positive 3 because the negatives cancel each other out.

Order of Operations

When solving mathematical expressions, it's important to follow a specific sequence of operations to achieve the correct result. This sequence is often remembered by the acronym PEMDAS:

• Parentheses: Resolve expressions inside parentheses first.
• Exponents: Calculate powers and roots next.
• Multiplication and Division: From left to right, as they appear in the expression.
• Addition and Subtraction: Also from left to right.

In our exercise, we are only dealing with multiplication, so we move from left to right. First, address any multiplication of negative numbers. Convert as necessary and then follow through with multiplying the rest of the sequence. The order of operations ensures consistency and correct results when dealing with complex expressions.

Integer Properties

Integers are whole numbers that include positive numbers, negative numbers, and zero. Under arithmetic operations such as addition, subtraction, and multiplication, integers exhibit certain properties:

• Closure Property: The product or sum of two integers is always an integer. For example, multiplying the integers 5 and -3 results in -15, which is also an integer.
• Commutative Property: Order does not matter when adding or multiplying. For multiplication, this means that changing the order of terms like (2)(3)(5) give the same product.
• Associative Property: Grouping of numbers does not affect the sum or product. For instance, (5 * (2 * 3)) yields the same result as ((5 * 2) * 3).
• Multiplicative Identity: Multiplying any integer by one gives the integer itself, such as 7 * 1 = 7.

In the provided solution, notice that the integers involved obey these properties. We simply use these properties to efficiently solve the exercise by rearranging and combining terms where needed.