Problem 33 - Beginning and Intermediate Algebra

Question

Multiply. $$ (-1)(2)(-3)(-5) $$

Step-by-Step Solution

Answer

The product is -30.

Step 1: Multiply the first two numbers

First, we'll multiply the first two numbers, which are \((-1)\) and \(2\). The calculation is \((-1) imes 2 = -2\).

Step 2: Multiply the result by the third number

Next, take the result from Step 1, which is \(-2\), and multiply it by the third number, \(-3\). The calculation is \((-2) imes (-3) = 6\).

Step 3: Multiply the result by the last number

Finally, take the result from Step 2, which is \(6\), and multiply it by the fourth number, \(-5\). This calculation is \(6 imes (-5) = -30\).

Key Concepts

Order of Operations

When solving mathematical expressions, it's crucial to follow the order of operations. This is a set of rules that tells us the correct sequence to evaluate a math expression, so everyone gets the same answer. The common acronym we use to remember these rules is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Each step in this order is necessary to simplify expressions and evaluate them correctly.

• Start by solving anything within parentheses first, though in some expressions they might not be present.
• Next, handle any exponents (or powers).
• After that, perform all multiplication and division as they appear from left to right.
• Finally, carry out any addition or subtraction, again from left to right.

In our exercise, we are dealing with multiplication only, so following the order of left to right ensures clarity and consistency. Multiplying each integer in sequence helps avoid mistakes and maintains the integrity of calculations.

Negative Numbers

Understanding negative numbers is fundamental in integer arithmetic. Negative numbers, which are less than zero, are represented with a minus sign (-) before them. They are used to represent losses, debts, temperatures below zero, and many other scenarios in real life. When multiplying numbers, knowing the behavior of these negative values is essential for accuracy.

• Multiplying a negative number by a positive number always results in a negative product.
• Multiplying two negative numbers results in a positive product because the two negative signs cancel each other out.
• If you multiply an odd number of negative numbers, the result will be negative.
• If you multiply an even number of negative numbers, you end up with a positive product.

In the example provided, first, a negative times a positive gives a negative. Then two negative numbers yield a positive when multiplied together. This pattern continues until the final answer is negative.

Integer Arithmetic

Integer arithmetic involves operations with whole numbers, which include positive numbers, negative numbers, and zero. Understanding integer arithmetic is key to tackling problems involving multiplication, division, addition, and subtraction. Here, we'll focus on multiplication, especially with integers.

• When multiplying integers, treat the operation similarly to regular multiplication, but pay close attention to the sign rules.
• Integers are closed under multiplication, which means that multiplying two integers always results in another integer.
• The product of integers follows commutativity and associativity properties, meaning the order in which numbers are multiplied doesn't change the result.

Notice how, step by step, we multiply the integers in the exercise. This illustrates how integer arithmetic and multiplication rules guide us through the problem. Understanding these basics ensures that any calculations with integers are handled properly, leading to correct outcomes.