Problem 56 - Elementary Algebra

Question

Perform the operations. $$ -4(-8) \div 16(-2) $$

Step-by-Step Solution

Answer

The result of the expression is -4.

Step 1: Simplify the Expression Inside Parentheses

First, we need to compute the expression involving the multiplication within the parentheses, \(-4(-8)\). Applying the rule that multiplying two negative numbers results in a positive number, we get \(-4 \cdot -8 = 32\).

Step 2: Simplify the Division

Now we have the expression \(32 \div 16 \times (-2)\). Apply the operation \(32 \div 16\), dividing 32 by 16 to get \(32 \div 16 = 2\).

Step 3: Multiply the Result by -2

Take the quotient from the previous step, which is 2, and multiply it by -2. According to the multiplication rule with negative numbers, a positive number multiplied by a negative number results in a negative product, so \(2 \times -2 = -4\).

Key Concepts

Multiplication of Negative Numbers

When we multiply two negative numbers, the result is always a positive number. This concept can seem counterintuitive, but it's a fundamental rule in algebra. Imagine you owe someone lunch points on two separate occasions. If each occasion represents a negative event, combining these events together puts you back in the positive.
The signs cancel each other out! So whenever you see an equation that requires multiplying two negative numbers, remember that the result will turn out positive.
In our exercise, multiplying

• -4 by -8

gives us their positive product, which is 32. This principle is consistent no matter how large or small the numbers are. It's like a negative debt being repaid, returning to a state of positivity.

Order of Operations

The order of operations is a set of rules to follow when simplifying mathematical expressions. It's crucial because it ensures consistency and correctness in solving problems. The order is often remembered by the acronym PEMDAS:

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

It's important to note that multiplication and division are performed at the same "tier" of importance, and should be calculated from left to right, as they appear in the problem. In our exercise, after addressing the parentheses

• we followed by handling the division, and only then proceeded to multiplication.

This strategy ensures we achieve the correct answer in an orderly and predictable manner. Mistakes happen when the order is disregarded, leading to incorrect results.

Division of Integers

Dividing integers can result in a whole number or a fraction. In algebra, it's vital to know whether you must round a result or express it as a fraction.
If both numbers are positive, or both are negative, the result is positive. However, if one number is negative and the other is positive, the quotient is negative.
Let's use our example:

• 32 divided by 16 produced a positive integer, 2.

We then multiplied 2 by a negative integer, -2, to get a final result of -4.
Always keep in mind the rules for sign determination:

• Two identical signs yield a positive result,
• while two different signs produce a negative result.

Understanding these guidelines ensures accurate division and helps make sense of more complex algebraic expressions.