Problem 15 - Elementary Algebra

Question

Simplify the following problems. $$ (-6)(-3) $$

Step-by-Step Solution

Answer

Answer: The product of (-6) times (-3) is +18.

Step 1: Identify the given expression

In this case, the given expression to simplify is: $$ (-6)(-3) $$

Step 2: Remember the rule for multiplying negative numbers

When we multiply two negative numbers, the result is a positive number.

Step 3: Multiply the numbers

Now we can multiply the absolute values of the numbers (ignoring the negative signs): $$ 6 \times 3 $$ Which equals 18.

Step 4: Determine the sign of the result

Since we are multiplying two negative numbers, the resulting product will be positive. So, the simplified expression is: $$ +18 $$

Key Concepts

Understanding Negative Numbers

Negative numbers are those that are less than zero. They are represented by a minus sign, such as \(-6\) or \(-3\). Understanding negative numbers is important as they are used to represent loss, debt, or a decrease in value.
In arithmetic, they behave differently from positive numbers. When you multiply or divide with negative numbers, you have to be careful with the signs.

• Adding a negative number is like subtracting its absolute value.
• Subtracting a negative number is like adding its absolute value.
• Multiplying two negative numbers results in a positive product.
• Dividing two negative numbers also results in a positive quotient.

Understanding these properties makes it easier to work with negative numbers in equations and expressions.

What is Absolute Value?

The absolute value of a number is the distance of the number from zero on a number line, regardless of its direction. It is always a non-negative number.
For example, both \(|6|\) and \(|-6|\) have an absolute value of 6.
Think of absolute value as ignoring the sign of a number. It's useful when solving mathematical problems involving negative numbers because it lets you focus on the size of the numbers without worrying about their signs.

• Absolute value is denoted by two vertical bars: \(|a|\).
• If \(a\) is positive or zero, then \(|a| = a\).
• If \(a\) is negative, then \(|a| = -a\).

Knowing how to calculate and interpret absolute values is crucial, especially when multiplying integers.

Multiplication Rules for Negative Integers

Multiplying integers, especially when they involve negative numbers, requires attention to the sign of the numbers involved.The key rule for multiplying negative integers is:

• The product of two negative integers is positive.
• The product of a positive integer and a negative integer is negative.
• If one of the numbers in a multiplication is zero, the result is zero.

To remember these rules, think of it in terms of direction on the number line: When two numbers with the same sign are multiplied (whether both are positive or both are negative), the direction remains the same, resulting in a positive number.
Conversely, multiplying numbers with different signs flips the direction, resulting in a negative number.
By understanding these multiplication rules, simplifying expressions like \((-6)(-3)\) to find that the answer is 18 becomes a straightforward process.