Problem 33 - Elementary Algebra

Question

Simplify. $$ (12) 3+(-2) 3 $$

Step-by-Step Solution

Answer

The simplified result is 30.

Step 1: Distribute the 3 to both terms

We start by distributing the multiplication of 3 to both terms inside the expression. The expression given is \((12) \, 3 + (-2) \, 3\). This means:\(12 \, 3 + (-2) \, 3\).

Step 2: Multiply each term individually

Calculate the multiplication for each term:1. \(12 \, \times \, 3 = 36\)2. \(-2 \, \times \, 3 = -6\)

Step 3: Combine the results

Now, we combine both results obtained in the previous step:\(36 + (-6)\)

Step 4: Simplify the expression

Finally, simplify the expression by performing the addition:\(36 + (-6) = 36 - 6 = 30\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, which helps us to simplify complex expressions. It allows us to distribute a single term over several terms inside a set of parentheses. This means multiplying the term outside the parentheses with each term inside it, one by one. This property is usually represented as:

• \( a(b + c) = ab + ac \).

In our exercise, the expression is \( (12) \, 3 + (-2) \, 3 \). Here, the number 3 is outside both instances, acting as the multiplier. We distribute 3 to both 12 and (-2), following the distributive property.
This step helps us transform the expression into a simpler format where each term is clearly separated by addition or subtraction. Understanding this concept lays the groundwork for many algebraic operations, enabling us to break down more complicated problems with ease.

Multiplication

Multiplication is the process of adding a number to itself a specified number of times. In algebra, multiplication plays a huge role, especially when simplifying expressions. In our exercise, multiplication was key to distributing the outside term to each inner term:

• First multiplication: \,\(12 \, \times \, 3 = 36\).
• Second multiplication: \,\((-2) \, \times \, 3 = -6\).

By performing these calculations, we can more effectively deal with each individual component of the original expression.
This step enables us to handle each part separately, making it easier to simplify as a whole. It is crucial for solving equations and inequalities, rendering the task less daunting by breaking it down into basic arithmetic.

Combining Like Terms

Combining like terms is an important process in algebra to simplify expressions further. Once each term has been distributed and multiplied, as in our exercise, the next step is to "combine like terms."
Like terms are terms that have identical variable parts or, as in this case, numerical values that are ready to be combined due to having no variable. In our example, after multiplying:

• We have 36 and -6 as the terms to be combined.

To combine them, we simply perform the addition or subtraction indicated:

• Calculate \(36 + (-6) = 30\).

Combining like terms simplifies the expression to its most compact form, making it clear and straightforward. This step is essential not only for arriving at the correct answer but also for maintaining mathematical simplicity and clarity throughout your work.