Problem 36 - Intermediate Algebra

Question

Perform the operations. See Example 3 . $$ -9(-1)(-3) $$

Step-by-Step Solution

Answer

The result is -27.

Step 1: Understand the Problem

We are asked to perform the multiplication of three negative numbers: \[-9 \times (-1) \times (-3)\] The goal is to find the product of these numbers by following the multiplication rules for negative numbers.

Step 2: Multiply the First Two Numbers

First, multiply the first two numbers: \[-9 \times (-1)\] According to the rules of multiplication, multiplying two negative numbers results in a positive number. Therefore, \[-9 \times (-1) = 9\]

Step 3: Multiply the Result by the Third Number

Now, take the result from Step 2 and multiply it by the third number: \[9 \times (-3)\] Since we are multiplying a positive number by a negative number, the result will be negative:\[9 \times (-3) = -27\]

Step 4: Conclusion

The final result of the given expression \(-9(-1)(-3)\) is the product from Step 3, which is \(-27\).

Key Concepts

Multiplication Rules

When multiplying numbers, one of the fundamental rules to remember is how the signs of the numbers affect the result.

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers results in a positive product because two negatives make a positive.
• Multiplying a positive number by a negative number, or vice versa, leads to a negative product.

To apply these rules, consider what will happen at each step in the example expression. Begin with any two of the numbers. For instance, multiplying the first two negative numbers leads to a positive result since both signs are negative. The outcome for two negatives such as -9 and -1 is positive 9.
The next step: multiply this positive 9 by the third negative number (-3), resulting in a negative product. Hence, 9 times -3 gives you -27.
These simple rules help simplify complex problems, especially when dealing with multiple negative factors.

Negative Numbers

Negative numbers can sometimes confuse students, particularly in multiplication. But the concepts become easier with clear rules. A negative number is any number less than zero, and it is symbolized by a minus sign (-).
When dealing with multiplication:

• Two negative numbers multiply to give a positive result, as their signs cancel each other out.
• If you multiply one negative and one positive number, the product remains negative, because the presence of a single negative sign dominates.

For example, the product of -9 and -1 is positive 9 because the negatives cancel out. However, when multiplying this positive 9 by -3, the product is negative 27.
Understanding how negative signs interact during multiplication can simplify solving algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Multiplying negative numbers within these expressions can often seem daunting, but the key is sequentially simplifying the terms using multiplication rules.
In our example, the expression \(-9(-1)(-3)\) is entirely based on numbers, but these same principles apply even when variables are added.

• Start by simplifying within the parentheses if they exist, just like in our worked example.
• Apply multiplication rules considering the signs of the numbers or coefficients.
• Recombine the simplified expressions using multiplication to solve the expression.

Breaking down each component step-by-step ensures accuracy and helps in understanding the algebraic processes involved.
With practice, dealing with negative numbers becomes second nature, even in more complicated algebraic expressions.