Problem 63 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 2 \sqrt{3}+7 \sqrt{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(9\sqrt{3}\).

Step 1: Identify Like Terms

In the expression \(2\sqrt{3} + 7\sqrt{3}\), both terms contain \(\sqrt{3}\) as a factor. This means they are like terms and can be combined. Unlike terms that have different radical parts cannot be combined, but here, the radical part is the same.

Step 2: Combine Like Terms

Because the terms are like terms, add their coefficients together. The coefficients are 2 and 7. Therefore, \(2 + 7 = 9\). So, \(2\sqrt{3} + 7\sqrt{3} = 9\sqrt{3}\).

Key Concepts

Like Terms

In algebra, like terms are terms that have the exact same variable part, which includes both the variable(s) itself and any exponent(s). When simplifying radical expressions, it’s essential to identify like terms so that they may be combined effectively. In the expression \(2\sqrt{3} + 7\sqrt{3}\), both terms include the same radical part, \(\sqrt{3}\), making them like terms.
Combining like terms is straightforward:

• Ensure the terms have identical variable parts.
• Add or subtract the coefficients, as you would with regular numbers.

When dealing with the expression \(2\sqrt{3} + 7\sqrt{3}\), you can directly add the coefficients (2 and 7) since the terms are like terms, resulting in \(9\sqrt{3}\). Recognizing and combining like terms simplifies expressions and forms the groundwork for solving more intricate algebraic equations.

Coefficients

Coefficients play a significant role in simplifying algebraic expressions, particularly those involving radicals. A coefficient is a numeric factor that multiplies a variable or, in the case of radicals, the radical part. In the expression \(2\sqrt{3} + 7\sqrt{3}\), the numbers 2 and 7 represent the coefficients of the terms.
When combining like terms in an expression, coefficients are what you manipulate to simplify the expression. Key steps include:

• Identifying the coefficients in each term of the expression.
• Ensuring the terms are indeed like terms.
• Adding or subtracting the coefficients as necessary to reach a simplified expression.

In our example, adding the coefficients, 2 and 7, gives you 9. This is multiplied by the shared radical part, resulting in \(9\sqrt{3}\). Understanding what coefficients are and how they function is crucial for handling various algebraic challenges.

Mathematics Education

Mathematics education involves teaching and learning mathematical concepts, including simplifying radical expressions. It focuses on developing a strong foundation in key mathematical principles so students can apply these concepts to solve problems in various contexts. Simplifying expressions is one of the many skills taught in mathematics education to build critical thinking and problem-solving abilities.
Educational strategies often include:

• Providing step-by-step solutions to show the process of solving complex problems.
• Emphasizing the understanding of core concepts, such as like terms and coefficients.
• Engaging students with problems that build their skills progressively.

Grasping the basics of simplifying radical expressions not only enhances algebraic proficiency but also lays the groundwork for future mathematical studies, encouraging students to approach problems logically and systematically.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. These expressions are fundamental in algebra and form a critical part of mathematics education. Simplifying algebraic expressions, such as those involving radicals, helps in cleaning up expressions to make them easier to work with.
Key elements of algebraic expressions:

• Variables - Symbols, often letters, that represent unknown values.
• Operations - Such as addition, subtraction, multiplication, and division.
• Numbers - Including constants and coefficients.

In the expression \(2\sqrt{3} + 7\sqrt{3}\), we see numbers acting as coefficients and \(\sqrt{3}\) serving as the radical part. Mastering the ability to simplify expressions like this one allows students to develop a more profound understanding of algebra, as it involves applying rules and operations governing these expressions. Understanding and simplifying algebraic expressions is crucial for advancing in mathematics.