Problem 14 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (-3)(-9) $$

Step-by-Step Solution

Answer

Answer: The product of (-3) multiplied by (-9) is 27.

Step 1: Identify the numbers

Observe the given math expression with two negative numbers that are being multiplied: $$ (-3)(-9) $$

Step 2: Apply the multiplication rule

Recall the rule when multiplying two negative numbers: the product will be a positive number. Hence, consider only the absolute values of the numbers and multiply them: $$ 3 \times 9 $$

Step 3: Perform the multiplication

Multiply the two numbers 3 and 9, resulting in: $$ 3 \times 9 = 27 $$

Step 4: Determine the sign of the result

Since both numbers being multiplied were negative, the result will be positive. The final answer is: $$ (-3)(-9) = 27 $$

Key Concepts

Multiplication Rule

When multiplying numbers, whether they are positive or negative, there are specific rules that help determine the sign of the product. The multiplication rule states that:

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers results in a positive product.
• Multiplying a positive number by a negative number results in a negative product.

To illustrate this rule in the given exercise, we have the expression \((-3)(-9)\).First, note that both numbers are negative. According to our multiplication rule, multiplying two negatives makes a positive.Therefore, the end result after applying the rule will be positive, which guides the rest of our calculations.

Negative Numbers

Negative numbers are numbers that are less than zero and are represented with a minus sign (-). They hold unique properties when it comes to various mathematical operations.

• Negative numbers are the opposite of positive numbers.
• When added to their positive counterparts, the result is zero (e.g., -3 + 3 = 0).
• In multiplication, an odd number of negative factors results in a negative product, while an even number of negative factors results in a positive product.

In the exercise \((-3)(-9)\), both factors are negative. Knowing that two negatives multiplied together result in a positive, we can initially set aside the negative signs, dealing with the absolute values for multiplication.

Absolute Values

Absolute value refers to the distance a number is from zero on a number line, without considering direction. In simple terms, the absolute value of a number is always positive.

• The absolute value of a positive number is the number itself (e.g., |3| = 3).
• The absolute value of a negative number is its positive counterpart (e.g., |-3| = 3).

For the expression \((-3)(-9)\), after understanding the multiplication of negatives, we use the absolute values for multiplication. We convert both -3 and -9 to their absolute values, 3 and 9, respectively. Thus, the calculation becomes simply multiplying these positive numbers: \( 3 \times 9 = 27 \), confirming that indeed, \((-3)(-9) \) results in 27.