Problem 26 - Algebra 1: Concepts and Skills

Question

Find the product. \(-2(-5)(7)\)

Step-by-Step Solution

Answer

The product of \( -2(-5)(7) \) is 70.

Step 1: Multiply the first two numbers

Multiply \( -2 \) and \( -5 \). According to the rule, the multiplication of two negative numbers gives a positive result. Thus, \( -2 \times -5 = 10 \).

Step 2: Multiply the result with the third number

Now, multiply the result from the first step (which is \( 10 \)) with the third number \( 7 \). The multiplication of a positive number and a positive number gives a positive result. Thus, \( 10 \times 7 = 70 \).

Key Concepts

Understanding Integer Properties

Integers are basic building blocks in math, composed of positive numbers, negative numbers, and zero. They play a key role in arithmetic operations such as addition, subtraction, multiplication, and division.

One important property of integers is closure under multiplication, which means when you multiply any two integers, the result is also an integer. For example, multiplying \(-2\) and \(-5\) results in \(-2 \times -5 = 10\), which is also an integer.

Additionally, integers obey the distributive, associative, and commutative properties for multiplication. These properties enable simpler and more efficient calculations while ensuring consistency in arithmetic operations. Let's emphasize the commutative property:

• Commutative property of multiplication: The order of multiplication does not affect the product, meaning \(a \times b = b \times a\).
• Associative property of multiplication: The way integers are grouped in multiplication does not change the product, so \(a \times (b \times c) = (a \times b) \times c\).

Understanding these properties helps you handle complex problems involving integers more effectively.

Multiplying Negative Numbers

When multiplying negative numbers, it's crucial to understand how their signs impact the result. The rule is simple yet significant: multiplying two negative numbers yields a positive product. This occurs because the two negatives "cancel out" due to the inherent property of opposites in mathematics.

To see why this works, consider \(-2 \times -5\). Both numbers have a negative sign, and multiplying them results in a positive product, \(10\). This principle is consistent across all negative integer multiplications.

The logic extends:

• A negative times a negative is a positive (e.g., \(-3 \times -4 = 12\))
• A positive times a negative remains a negative (e.g., \(3 \times -4 = -12\))
• A negative times a positive is also a negative (e.g., \(-3 \times 4 = -12\))

Mastering this rule allows for accurate and confident multiplication of negative numbers across varied mathematical scenarios.

Positive and Negative Numbers in Multiplication

Combining positive and negative numbers in multiplication can initially seem puzzling. However, recognizing how their signs govern the final result is essential to mastering integer multiplication.

Here’s what you need to know:

• When multiplying a positive number by another positive number, the outcome is always positive, like \(7 \times 3 = 21\).
• Multiplying a negative number by a positive number results in a negative product, as the negative dominates (e.g., \(-5 \times 4 = -20\)).
• Conversely, a positive multiplied by a negative also yields a negative outcome (e.g., \(3 \times -8 = -24\)).

Understanding these interactions helps anticipate the sign of any product. This foresight is invaluable in simplifying complex calculations or verifying results, ensuring clarity and accuracy in math problem-solving.

Remember, only two negative integers multiply to form a positive product, as explained earlier—understanding these rules ensures a grasp of the fundamental operations with integers.