Problem 25 - 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. $$(-50)(15)(-2)-(-4)(17)(25)$$

Step-by-Step Solution

Answer

The simplified expression is 3200.

Step 1: Simplify Each Group of Terms

First, simplify the expression by multiplying the numbers within each group of parentheses separately. Begin by calculating \((-50)(15)(-2)\) and \((-4)(17)(25)\). For the \((-50)(15)(-2)\): Multiply \((-50)\) by \(15\) to get \(-750\). Then multiply the result by \((-2)\) to get \(1500\) since multiplying two negative numbers yields a positive number. For the second group:Multiply \((-4)\) by \(17\) to get \(-68\).Then multiply the result by \(25\) to get \(-1700\).

Step 2: Substitute and Combine

Replace the results back into the expression from step 1, giving:\(1500 - (-1700)\).Notice that the subtraction of a negative number is equivalent to adding a positive number due to the properties of integers.

Step 3: Simplify the Final Expression

Add \(1500\) and \(1700\) (since subtracting a negative is adding a positive):\(1500 + 1700 = 3200\).

Key Concepts

Simplification

Simplification is all about making complex numerical expressions easier to work with. In mathematics, it involves reducing expressions to their simplest form so that calculations become more straightforward. When presented with a problem like multiplying several numbers, it's important to handle and simplify the terms step by step.

• Start by addressing the operations within parentheses or any grouped terms. This is in accordance with the order of operations.
• Next, perform any multiplication or division operations in sequence.
• Finally, handle the addition and subtraction to arrive at the simplest form of the expression.

These techniques reduce the likelihood of errors and make the problem easier to understand. Efficient simplification relies heavily on recognizing and using mathematical properties.

Properties of Integers

The properties of integers are foundational rules that help in simplifying mathematical expressions. These properties include commutative, associative, and distributive properties. Let's explore them.

• The commutative property of multiplication allows numbers to be multiplied in any order without changing the result. For example, \((-50) \times 15 \times (-2)\) can be calculated in any order due to this property.
• The associative property deals with grouping. It suggests that the way numbers are grouped when multiplied doesn’t affect the product: i.e., \((a \times b) \times c = a \times (b \times c)\).
• Lastly, the distributive property links addition with multiplication: \(a \times (b + c) = a \times b + a \times c\). This helps in breaking down expressions into more manageable parts.

Using these properties simplifies problems, making calculations more manageable and errors less likely.

Multiplication of Integers

Multiplying integers follows specific rules that must be understood for accurate calculations. Here's what you need to know:

• When multiplying two positive integers, the result is always positive. For instance, \(3 \times 5 = 15\).
• When multiplying a positive integer with a negative integer, the result is always negative. For example, \(-4 \times 6 = -24\).
• Multiplying two negative integers results in a positive integer, a counterintuitive but essential rule: \((-7) \times (-8) = 56\).

These rules are crucial in various mathematical computations, ensuring that multiplication is performed accurately regardless of the signs involved. Understanding them avoids confusion and miscalculations.

Negative Numbers

Negative numbers can often be tricky, but they're simply numbers less than zero on the number line. Understanding their properties and how they interact with other integers in operations are critical skills.

• Subtracting a negative number is equivalent to adding the positive version of that number. This concept is used when converting \(a - (-b)\) into \(a + b\).
• A negative times a positive is negative, as mentioned, and two negatives multiplied yield a positive.
• When comparing, note that a negative number is always less than a positive number.

Operations involving negative numbers follow these rules and ensure that mathematical expressions are simplified and solved correctly. Mastery of these concepts makes working with integers much less daunting.