Problem 25 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ (-50)(15)(-2)-(-4)(17)(25) $$

Step-by-Step Solution

Answer

3200

Step 1: Simplify Each Term Separately

First, we simplify each product within the expression: For \[(-50)(15)(-2),\]we multiply these three numbers step-by-step. Start with \[(-50)(15) = -750.\]Then, multiply the result by \[(-2):\]\[-750 \times (-2) = 1500.\]For \[(-4)(17)(25),\]we begin by finding \[(-4)(25) = -100.\]Then multiply by \[17:\]\[(-100)(17) = -1700.\]

Step 2: Perform the Subtraction

Now, we substitute these simplified terms back into the expression. The problem becomes:\[1500 - (-1700).\]Remember that subtracting a negative number is equivalent to adding the positive of that number. Thus, the expression simplifies to:\[1500 + 1700.\]

Step 3: Calculate the Final Result

Add \[1500 + 1700\] to find the final result:\[1500 + 1700 = 3200.\]

Key Concepts

Properties of operations

Understanding the properties of operations can make simplifying numerical expressions easier and more efficient. Key properties that play a crucial role in arithmetic include:

• Associative Property: This property applies to addition and multiplication, allowing you to group numbers differently without changing the result. For instance, in multiplication, \((-50)(15)(-2)\) can be regrouped to simplify in stages. Initially solving \((-50)(15)\) as \-750\, then \(-750 \times (-2) = 1500\).
• Commutative Property: This property suggests the order of addition or multiplication does not affect the outcome. Although it wasn't directly needed in this exercise, it's handy, for example, shifting factors in \((-4)(17)(25)\) to find an easier multiplication path.
• Distributive Property: Combines multiplication over addition or subtraction. It wasn't directly applied here, but it's fundamental when dealing with expressions that combine these operations, simplifying before multiplying further.

Recognizing when and which properties to use helps cut down complex tasks into simplified operations.

Negative numbers

Working with negative numbers can seem tricky at first, but it's all about understanding their behavior with different operations. Here are some points to consider:

• Multiplying Negatives: When multiplying two negative numbers, the result is a positive number. This is because a negative indicates a direction, and doubling the negative reverses it back to positive. In our exercise, \([-750 \times (-2) = 1500]\) demonstrates this principle effectively.
• Negative with Subtraction: Subtracting negative numbers is equivalent to adding their positive counterparts. Recall from the exercise \([1500 - (-1700)]\) turns into \([1500 + 1700]\).

These rules simplify calculations by reducing potential mistakes, making negative numbers less daunting.

Order of operations

The order of operations is a fundamental principle that dictates the sequence in which operations should be performed to achieve the correct result. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it guides the simplification path:

• Multiplication Before Addition: In the original problem, completing all multiplication tasks within each term first helped simplify the expression systematically. This involved finding \(-750\) from \((-50)(15)\) before incorporating \(-2\).
• Handling Subtraction: The expression \([1500 - (-1700)]\) is simplified by recognizing the final step as adding a positive, emphasizing the role "subtraction of negatives equals addition."

By correctly applying the order of operations, complex expressions are resolved accurately and efficiently, preventing computation errors that arise from arbitrary operation orders.