Problem 16 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (-5)(-2) $$

Step-by-Step Solution

Answer

Answer: The result of multiplying $$-5$$ and $$-2$$ is 10.

Step 1: Multiply the numbers

To find the value of the given expression, we just need to multiply the two numbers together. So, we multiply $$-5$$ and $$-2$$ like this: $$ (-5)(-2) = 5 \times 2 $$

Step 2: Get the result

Now that we have multiplied the two numbers, we can find the result: $$ 5 \times 2 = 10 $$ The value of the expression $$ (-5)(-2) $$ is 10.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. You can think of them as numbers going left of zero on a number line. They are often used to represent values below a standard quantity, such as temperatures below freezing or debts in finance.

When dealing with negative numbers, pay attention to their signs. Signs determine whether numbers are considered positive (without a minus sign) or negative (with a minus sign).

• Negative numbers are often expressed with a "-" before the number, like -5.
• They are less than positive numbers.

One important thing about negative numbers is they often change how mathematical operations work. For example, subtracting a negative number or multiplying two negative numbers leads to unique outcomes. Understanding these properties helps in solving algebraic problems effectively.

Rules for Multiplying Numbers

Multiplying numbers follows some simple rules, especially when negative numbers come into play. These rules help us predict the sign of the answer without even doing the full multiplication.

Here are a few multiplication rules to keep in mind:

• Positive × Positive = Positive: When you multiply two positive numbers, like 2 × 3, you get a positive number: 6.
• Negative × Positive = Negative: Multiplying a negative number by a positive number, like -2 × 3, results in a negative number: -6.
• Positive × Negative = Negative: This situation is similar to the above. If you reverse the order and multiply a positive number by a negative number, the rule stays the same: 2 × -3 = -6.
• Negative × Negative = Positive: This is perhaps the most surprising rule. When you multiply two negative numbers, like -2 × -3, the negatives "cancel out," and you get a positive number: 6.

Understanding these rules is crucial when approaching algebraic expressions because the placement of negative signs can significantly change the outcome.

Algebraic Expressions Involving Multiplication

Algebraic expressions often combine numbers, variables, and operations. When you encounter algebraic expressions that include multiplication, it is important to understand the role each number and symbol plays.

An algebraic expression might look like this: \(-3x\), where \(-3\) is multiplied by a variable \(x\).

When working with expressions:

• Combined Elements: Expressions like \(ab\) imply multiplication between the elements \(a\) and \(b\).
• Handling Negative Coefficients: Understanding how to manage negative coefficients is important. For instance, \( -3x\) is read as \(-3\ times x\). Keeping track of multiplication rules helps when simplifying or evaluating these expressions.
• Expanding Expressions: Sometimes, expressions need expanding using the distributive property, like turning \(a(b + c)\) into \(ab + ac\).

Using these principles, you can solve and simplify algebraic expressions efficiently. Keeping multiplication rules in mind ensures you accurately handle signs, leading to correct results.