Problem 20 - 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. $$(14)(25)(-13)(4)$$

Step-by-Step Solution

Answer

The simplified expression is \(-18200\).

Step 1: Group the Positive and Negative Numbers

First, identify and group the positive and negative numbers. In the equation \((14)(25)(-13)(4)\), we have 14, 25, and 4 as positive numbers and -13 as a negative number.

Step 2: Compute the Product of the Positive Numbers

Multiply all the positive numbers: \(14 \times 25 \times 4\).First, calculate:\(14 \times 25 = 350\).Then multiply the result by 4:\(350 \times 4 = 1400\).

Step 3: Introduce the Negative Number

Now multiply the product of the positive numbers by the negative number:\(1400 \times (-13)\).

Step 4: Solve the Multiplication with the Negative Number

Perform the multiplication:\(1400 \times (-13) = -18200\).Since multiplication by a negative number changes the sign, the result is negative.

Key Concepts

Properties of Multiplication

To simplify numerical expressions, understanding the properties of multiplication is crucial. These properties help to reorder and group numbers in a way that makes calculations much simpler. Here are key properties:

• Commutative Property: This property states that the order in which you multiply numbers does not affect the product. For instance, \(a \times b = b \times a\). This property is particularly useful when trying to simplify expressions by grouping easier numbers to multiply first.
• Associative Property: According to this property, how you group the numbers does not affect the product. For example, \((a \times b) \times c = a \times (b \times c)\). This allows you to multiply numbers together in the grouping that is simplest to compute.
• Distributive Property: Though not explicitly used in our numerical expression, this property is also helpful in breaking down problems: \(a \times (b + c) = a \times b + a \times c\).

In our problem, using these properties made it easier to first multiply the positive numbers (14, 25, and 4) before introducing the negative number, which streamlined the calculation process.

Multiplication of Positive and Negative Numbers

Understanding how multiplication of positive and negative numbers works is imperative for simplifying expressions, especially when they include both. When multiplying numbers, the product's sign depends on the combination of those numbers. Here are some rules:

• Positive × Positive = Positive: Just like multiplying any ordinary numbers, the product remains positive.
• Positive × Negative = Negative: A positive and a negative number multiply to yield a negative product. Think of this as the positive number losing its positivity due to the negative.
• Negative × Negative = Positive: When two negative numbers are multiplied, the negatives cancel each other out, resulting in a positive product.

In the original expression \(14 \times 25 \times (-13) \times 4\), once we multiplied all the positive numbers together, our intermediate product \(1400\) needed to be multiplied by \(-13\). The multiplication followed the positive × negative rule, resulting in a negative (\(-18200\)).

Step by Step Problem Solving

Breaking down a complex multiplication problem into steps enhances clarity and reduces errors. Let's go over our example step by step to ensure understanding:First, identify and group similar types of numbers, such as positive and negative numbers, to focus on their individual properties. In our expression, it helped to treat positive numbers together and separate the negative one.

• Step 1: Group and Multiply Positive Numbers: Start with simpler multiplication by handling positive numbers together. Multiply \(14 \times 25\) yielding \(350\), and continue by multiplying that result with \(4\) to get \(1400\).
• Step 2: Integrate Negative Multiplication: After obtaining a product from the positive numbers, introduce the negative number. Multiply \(1400\) by \(-13\). This calculation gives \(-18200\) because the multiplication of positive and negative results in a negative.

By dividing the expression into manageable segments, you address each part effectively, ensuring a streamlined and successful calculation.