Problem 96 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -5(-9)(-4 n) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-180n\).

Step 1: Identify Parts of the Expression

The expression given is \(-5(-9)(-4n)\). It contains three terms: \(-5\), \(-9\), and \(-4n\), which are connected by multiplication.

Step 2: Simplify Constants

First, multiply the constants \(-5\) and \(-9\). The calculation is:\[-5 \times -9 = 45\]This yields 45.

Step 3: Multiply with the Variable Term

Now, take the result from Step 2 and multiply it by the remaining term, \(-4n\). Using our result:\[45 \times (-4n) = -180n\]

Step 4: Combine Results

After performing the multiplication, the simplified form of the expression is: \(-180n\).

Key Concepts

Multiplication with Variables

When dealing with algebra, you often encounter variables together with numbers. Variables represent unknown values, typically shown as letters like \( n \) in algebraic expressions. Multiplication with these variables involves treating the variable as a factor just like numbers. For example, if you have an expression like \(-4n\), you're essentially looking at \(-4 \times n\).

Here, principles of multiplication apply in the same way as they do with regular numbers. That means:

• Multiplying a variable by a constant simply scales the variable's quantity. For instance, multiplying \( n \) by 3 becomes 3\( n \).
• Negative numbers play a crucial role. Multiplying a negative number by a positive number results in a negative product. Similarly, multiplying two negative numbers produces a positive product.

In the example of \(-5(-9)(-4n)\), all terms are multiplying together. Constant multiplication is done first, and then the result is multiplied by the term that includes the variable.

Simplifying Expressions

The goal of simplifying expressions in algebra is to make them as straightforward as possible. This involves condensing the expression by combining like terms, performing arithmetic operations, and following mathematical conventions in a step-by-step process.

To simplify, follow these steps:

• Identify and isolate parts of the expression. In \(-5(-9)(-4n)\), you initially separate out the numeric constants and variable terms.
• Perform operations on constants first. As shown in the example, multiplying \(-5\) and \(-9\) gives \(45\), because the product of two negative numbers is positive.
• Next, apply this numeric sum to any variables within the expression. In our case, the result of 45 gets multiplied by \(-4n\), resulting in \(-180n\). The negative sign from \(-4\) remains as it affects the overall product.

Through these steps, the expression becomes more manageable and easier to work with or understand in further calculations.

Basic Algebra Operations

Basic algebra operations involve addition, subtraction, multiplication, and division. Each of these helps in manipulating expressions and solving equations.

- **Addition and Subtraction:** Used to combine like terms. For example, \(2n + 3n\) results in \(5n\) because both terms have the same variable part.
- **Multiplication:** The main focus in our exercise involves using multiplication, particularly when variables are included. You multiply coefficients (the numbers in front of variables) and then append the variable, as seen with \(-4n\) leading to \(-180n\).
- **Division:** This is used to separate terms or simplify expressions further but was not needed in this specific example.

By mastering these basic operations, you can tackle more complex algebraic problems. Remember that the order in which operations are performed, often influenced by negative and positive signs as well as brackets, is essential for correct solutions. The operation of multiplication, especially, sets the stage for scaling variables and working with algebraic expressions efficiently.