Problem 117 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ a+a+a $$

Step-by-Step Solution

Answer

The expression \( a + a + a \) simplifies to \( 3a \).

Step 1: Identify the Expression

The given expression is \( a + a + a \), which consists of three similar terms, all being the variable \( a \).

Step 2: Apply the Rule of Addition

Since all the terms are similar, we can add them together. In algebra, adding three \( a\)s gives us \( 3 \times a \) or simply \( 3a \).

Step 3: Write the Simplified Expression

The simplified form of \( a + a + a \) is \( 3a \). This step involves writing the expression in its simplest form, combining all similar terms.

Key Concepts

Understanding Simplification

Simplification in algebra is the process of making an expression easier to understand or work with by reducing it to its simpler form.
When we look at an expression, we aim to combine all possible parts, making it as compact as possible. This involves a keen eye for similar terms and mathematical operations that can be combined.

• For instance, in the expression \( a + a + a \), the process of simplification recognizes multiple occurrences of the same term and compacts them into a cleaner form.
• The main goal is to express the mathematical idea clearly and concisely, often by reducing redundant parts.

Simplifying expressions is essential because it helps in solving equations more efficiently and can reveal insights or patterns, making the math process smoother.

The Role of Addition in Expressions

Addition is one of the fundamental operations in algebra.
When it comes to expressions with like terms, addition allows us to combine these terms into a single term, thereby simplifying the expression.

• For the expression \( a + a + a \), addition involves counting how many times the variable \( a \) appears, which is three in this case.
• This counting allows you to rewrite the expression as a single term with a coefficient, such as \( 3a \). This shows the total quantity or sum of the variables involved.

Addition in algebra can be thought of as summing units of the same entity. When this is done correctly, it leads to a cleaner, more manageable expression that retains the same meaning and value as the original expression.

Identifying and Simplifying Like Terms

Like terms in algebra are terms that have exactly the same variable part. This means they are multiplicative combinations of the same variables raised to the same powers. They are essential to recognize in simplifying expressions.

• In the example \( a + a + a \), each term is a 'like term' because every one of them contains the variable \( a \) with the same degree, which is 1.
• To simplify expressions with like terms, you add the coefficients together. Here, since there are three \( a \)s, the expression can be condensed to \( 3a \).

Recognizing like terms quickly in expressions is crucial because it reduces the complexity of algebraic manipulations and allows for effective calculation and problem-solving. Once you can identify like terms, simplification becomes straightforward, making subsequent algebraic operations easier to perform.