Problem 26 - 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. $$ (2)(17)(-5)-(4)(13)(-25) $$

Step-by-Step Solution

Answer

The simplified expression is 1130.

Step 1: Understand the Expression

The given expression is \((2)(17)(-5) - (4)(13)(-25)\). We need to simplify each term separately before subtraction.

Step 2: Simplify the First Term

Simplify \((2)(17)(-5)\) by multiplying the numbers sequentially. Start with \(2 \times 17 = 34\), and then multiply \(34 \times (-5) = -170\). So, \((2)(17)(-5) = -170\).

Step 3: Simplify the Second Term

Simplify \((4)(13)(-25)\) by multiplying the numbers sequentially. Start with \(4 \times 13 = 52\), and then multiply \(52 \times (-25) = -1300\). So, \((4)(13)(-25) = -1300\).

Step 4: Subtract the Simplified Terms

Now subtract the results of the two terms: \(-170 - (-1300)\). This is equivalent to \(-170 + 1300\).

Step 5: Calculate the Result

Calculate \(-170 + 1300\), which gives \(1130\). Therefore, the simplified numerical expression is \(1130\).

Key Concepts

Properties of Multiplication

When dealing with multiplication, it's important to understand the properties that govern it, as they can simplify computations. The main properties include:

• Commutative Property: This states that switching the order of numbers in a multiplication equation does not change the result. For example, \( a \times b = b \times a \).
• Associative Property: This highlights that, when multiplying three or more numbers, the grouping does not affect the product. For example, \( (a \times b) \times c = a \times (b \times c) \).
• Distributive Property: This allows multiplication to be distributed over addition or subtraction. For instance, \( a \times (b + c) = a \times b + a \times c \).

By applying these properties, complex multiplication expressions can be simplified, making them easier to handle. In the exercise, we use these insights for sequential multiplication of terms, which, combined with understanding signs, helps simplify the expression efficiently.

Sequential Multiplication

Sequential multiplication refers to the process of multiplying numbers in a specific sequence. This step-by-step approach helps manage complex calculations smoothly:

• Start with Two Numbers: Begin by multiplying the first two numbers in the sequence. Like for \( (2)(17)(-5) \), you start with \( 2 \times 17 = 34 \).
• Proceed with the Resultant Product: Take the product from the first step and multiply it with the next number in sequence. From the previous step, multiply \( 34 \times (-5) = -170 \).
• Repeat as Necessary: Continue this process if more numbers are present, focusing on two at a time.

Each multiplication operation should consider the proper handling of positive and negative signs to avoid errors. The result of the sequence provides the simplified term needed for further operations.

Subtracting Negative Numbers

Understanding how to subtract negative numbers is crucial, especially when simplifying expressions. The concept is straightforward if you remember a few key rules:

• Negative Minus Negative: When subtracting a negative number, the operation turns into addition. For instance, the expression \(-a - (-b)\) becomes \(-a + b\).
• Visualize on the Number Line: Consider the number line: moving left suggests subtraction, while moving right suggests addition.
• Example: In the calculation \(-170 - (-1300)\), it simplifies to \(-170 + 1300 \), which ultimately provides \(1130\).

Grasping these key ideas eliminates the confusion around double negatives in arithmetic, allowing smoother calculations and accurate results.