Problem 28 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3 a^{2}+6 a^{2}+2 a^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(3a^{2} + 6a^{2} + 2a^{2}\) Answer: \(11a^{2}\)

Step 1: Identify Like Terms

In the expression, we can see that all the terms are like terms as they are all multiples of \(a^{2}\): $$ 3 a^{2}+6 a^{2}+2 a^{2} $$

Step 2: Combine Like terms

Next, we add the coefficients of the like terms and retain the base and exponent as it is: $$ (3 + 6 + 2) a^{2} $$

Step 3: Compute the final result

Add the sum of the coefficients and write their sum in front of the base and exponent: $$ 11 a^{2} $$ The simplified expression is \(11a^{2}\).

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have identical variables and exponents. It's essential to properly identify like terms when simplifying expressions because they are the only ones that can be combined. For example, in the expression \(3a^2 + 6a^2 + 2a^2\), all terms contain the same variable \(a^2\), making them like terms. Here are some key points to remember about like terms:

• Like terms must have the exact same variable and exponent. For example, \(4x^2\) and \(5x^2\) are like terms, but \(4x\) and \(4x^2\) are not.
• Coefficients can be different. It's the combination of the variable and exponent that matters.
• Like terms can include constants. For example, \(3\) and \(5\) are like terms, but they usually appear separately as their own group.

Finding and understanding like terms is the bedrock of simplifying algebraic expressions. It helps you spot which parts of the expression can be combined, making it shorter and easier to interpret.

The Art of Combining Like Terms

Once you identify the like terms in an algebraic expression, the next step is to combine them to simplify the expression further. Combining like terms involves two primary steps: adding or subtracting their coefficients and retaining the variables with their exponents.In our expression \(3a^2 + 6a^2 + 2a^2\), all terms are like terms because they involve \(a^2\). You simply add the coefficients:

• Add: \(3 + 6 + 2 = 11\)
• Retain the variable and its exponent: \(11a^2\)

Here are some helpful tips when combining like terms:

• Ensure that terms truly are like terms before combining them. Double-check the variables and their exponents.
• Consider negative coefficients and subtraction as operations in the combination process.
• Use parentheses to help group like terms, especially in more complex expressions.

By mastering the art of combining like terms, you can make expressions more compact and manageable, which is crucial for solving equations efficiently.

Algebra Basics Simplified

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. At its core, it is about finding the unknown values represented by these symbols. Being comfortable with algebra basics, like understanding terms, expressions, and how to simplify them, sets the foundation for more complex problem solving.Some fundamental elements include:

• **Terms**: A single mathematical expression could be a number, a variable, or numbers and variables multiplied together. Example: \(3a\), \(4b\), or \(5c^2\).
• **Expressions**: A combination of terms via addition or subtraction. Example: \(3a + 4b - 5c\).
• **Coefficients**: Numbers that multiply a variable within a term. In \(4x\), \(4\) is the coefficient.
• **Simplifying**: Reducing an algebraic expression by combining like terms and applying mathematical operations.

Understanding these basics helps in recognizing patterns and simplifying or solving equations efficiently. It makes algebra seem less like a foreign language and more like a logical puzzle.