Problem 25 - Elementary Algebra

Question

Simplify the following problems. $$ 0-6(-4)(-2) $$

Step-by-Step Solution

Answer

Question: Simplify the numerical expression $$0 - (-6(-4))(-4(-2))$$. Answer: The simplified expression is $$-192$$.

Step 1: 1. Identify the order of operations in the expression

The correct order of operations is Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this expression, we have subtraction and multiplication. We should start with the multiplication first.

Step 2: 2. Perform the multiplication

We have two multiplications in the expression: $$-6(-4)$$ and $$(-4)(-2)$$. Let's proceed with these calculations one by one. 1. $$-6(-4) = 24$$ (a negative number multiplied by a negative number results in a positive number) 2. $$(-4)(-2) = 8$$ (again, a negative number multiplied by a negative number results in a positive number) Now our expression becomes: $$0 - 24(8)$$

Step 3: 3. Perform the remaining multiplication

Next, we can proceed with multiplying 24 and 8. $$24(8) = 192$$ Now the expression is: $$0 - 192$$

Step 4: 4. Perform the subtraction

Finally, we need to subtract 192 from 0. $$0 - 192 = -192$$ The simplified result of the expression is $$-192$$.

Key Concepts

Order of Operations

Understanding the order of operations is essential when simplifying algebraic expressions. It provides a clear guideline on how to approach the problem, ensuring that everyone arrives at the same solution. Often remembered by the acronym **PEMDAS** or **BODMAS**, these rules dictate that you should first solve operations inside Parentheses/Brackets, then handle any Exponents/Orders. Follow this by performing any Multiplication and Division from left to right, and finally, address Addition and Subtraction, also from left to right.

In the exercise, the expression given includes both subtraction and multiplication. According to the order of operations, multiplication takes priority over subtraction. This means before dealing with the subtraction aspect, we must first complete all multiplication tasks in the expression.

• Begin with calculations involving parentheses (if any exist).
• Move to multiplying numbers next.
• Finally, perform subtraction to conclude the operation.

By adhering to this sequence, computational errors are minimized, and the equation is handled systematically, leading to the correct result.

Multiplication Rules

Multiplying integers, particularly negative numbers, can sometimes be tricky. It is crucial to remember the rules pertaining to the signs of numbers in multiplication. Here’s a quick refresher:

• When multiplying two positive numbers, the result is always positive.
• When multiplying a positive number by a negative number (or vice versa), the result is always negative.
• When two negative numbers are multiplied, the result is positive.

In our exercise, multiplications involve pairs of negative numbers, specifically \(-6(-4)\) and \((-4)(-2)\). Both calculations follow the rule where a negative times a negative equals a positive number. Therefore, the results are 24 and 8, respectively. This conversion of what could have been confusing sub-calculations into straightforward positives makes simplifying the expression manageable.

After handling the separate multiplications, we then multiply what we got, 24, by another positive number, 8, to maintain the sequence of operations. Adhering to these multiplication rules prevents mistakes from arising out of misunderstanding or misapplying sign rules.

Subtraction in Algebra

Subtraction, the final step in our expression simplification, is not merely taking one number away from another. It involves understanding that subtraction is fundamentally about adding the inverse. This means that subtracting a number is equivalent to adding its negative.

In our expression, after executing the multiplication steps, we are left with \(0 - 192\). Here, 192 is positive, and subtracting it from 0 is essentially like adding \(-192\). This approach can help clarify what the operation really entails:

• View subtraction as adding a negative number.
• This shifts the numerical value in the opposite direction on the number line.

The outcome, in this case, is \(-192\), demonstrating how, through this lens, negative results emerge naturally when subtracting a number from zero. By recognizing subtraction as such, algebraic manipulations become not only more intuitive, but they also build a foundation for more complex arithmetic operations.