Problem 19 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$(25)(-13)(4)$$

Step-by-Step Solution

Answer

The simplified expression is -1300.

Step 1: Simplify the Positive Product

First, multiply the positive numbers together: \( 25 \times 4 = 100 \).

Step 2: Incorporate the Negative Number

Next, multiply the result from Step 1 by \(-13\): \( 100 \times (-13) = -1300 \).

Key Concepts

Multiplication Properties

Understanding multiplication properties is essential for simplifying numerical expressions effectively. These properties provide techniques that make complex calculations easier and faster.

• Commutative Property: This property states that the order of numbers does not affect the product. In mathematical terms: \( a \times b = b \times a \). For example, \( 25 \times 4 \) is the same as \( 4 \times 25 \), which is useful in rearranging numbers for simpler multiplication.
• Associative Property: This outlines that the way numbers are grouped in multiplication does not change the product. For example, \((a \times b) \times c = a \times (b \times c)\). This allows you to group multiplying numbers in a way that makes the calculation easier.
• Distributive Property: This property connects addition and multiplication, typically used to simplify expressions. However, it can aid in understanding how numbers interact when multiplied together in parts: \( a \times (b + c) = (a \times b) + (a \times c) \).

By using these properties strategically, breaking down complex expressions becomes more manageable.

Negative Numbers

Negative numbers can initially seem challenging, but they follow straightforward rules that make calculations easier to navigate. Understanding how negative numbers interact in multiplication is crucial.

• Negative Sign in Multiplication: When a positive number is multiplied by a negative number, the result is always negative. For example, \(100 \times (-13) = -1300\). The product takes the sign of the negative number.
• Two Negative Numbers: If both numbers are negative in multiplication, the product will be positive. This is because the negative signs cancel each other out. For example, \((-a) \times (-b) = a \times b)\).

Grasping these rules will help you to gain confidence in handling expressions involving negative numbers.

Simplification Steps

To tackle numerical expressions efficiently, breaking down the problem into smaller steps is useful. This methodical approach makes it easier to arrive at the correct answer.

• Identify Like Operations: Start by grouping numbers with similar operations. In the given example, begin with the positive numbers: \(25 \times 4 = 100\).
• Process Step-by-Step: Implement multiplication sequentially. After finding the positive product, incorporate the negative number: \(100 \times (-13) = -1300\).
• Double-check Your Work: Reassessing each step ensures accuracy. Confirm each multiplication stage follows specified rules, such as proper handling of negative signs.

Simplifying expressions systematically not only ensures precision but also builds your mathematical intuition over time.