Problem 171 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c} $$

Step-by-Step Solution

Answer

\( \sqrt{3c} \)

Step 1: Identify Like Terms

Recognize that each term includes the same radical \( \sqrt{3c} \). This means the terms \( 8 \sqrt{3c}, 2 \sqrt{3c}, \text{and} \-9 \sqrt{3c} \) are like terms and can be combined.

Step 2: Combine Like Terms

Add and subtract the coefficients of the like terms: \( 8 + 2 - 9 \). Start by adding 8 and 2 to get 10. Then subtract 9 to get 1.

Step 3: Simplify the Expression

Multiply the final coefficient by the common radical: \( 1 \cdot \sqrt{3c} \). The simplified form of the expression is \sqrt{3c}.

Key Concepts

like terms

When simplifying mathematical expressions, identifying like terms is a crucial skill.
Like terms are terms that contain the same variable or radical part. This means they can be combined together.

• For example, in the expression \(8 \sqrt{3c} + 2 \sqrt{3c} - 9 \sqrt{3c}\), all the terms contain the same radical \( \sqrt{3c} \).
• This makes them like terms and allows us to combine them by working with their coefficients.

Recognizing like terms helps in simplifying expressions, making calculations easier, and reducing the complexity of algebraic operations.
Look for terms with identical variable components or radicals next time you work with expressions!

combining radicals

The process of combining radicals involves adding or subtracting the coefficients of like terms.
Here’s how you can do it step-by-step:

• First, identify the like terms, which we already confirmed as \( 8 \sqrt{3c}, 2 \sqrt{3c}, \text{and} \ -9 \sqrt{3c} \).
• Then, add or subtract their coefficients: \( 8 + 2 - 9 \).
• When combined, \ 8 + 2 = 10 \, and then \ 10 - 9 = 1 \.
• The result is \ 1 \sqrt{3c} \.

By combining the coefficients, we simplify the expression from complex multiple terms to a single term.
It’s as simple as handling basic arithmetic operations but focusing on the coefficients of identical radical parts.

coefficients

In algebra, a coefficient is the numerical part of a term that contains a variable or radical.
For instance, in the term \( 8 \sqrt{3c} \), the number 8 is the coefficient.
Understanding and working with coefficients is essential when simplifying expressions because:

• They determine how many units of the variable or radical we are dealing with.
• When combining like terms, it's the coefficients that are added or subtracted, not the variable or radical part itself.

For example, in the expression \( 8 \sqrt{3c} + 2 \sqrt{3c} - 9 \sqrt{3c} \), the coefficients (8, 2, -9) are combined separately.
The final expression \( 1 \sqrt{3c} \) simplifies by reducing these numbers, keeping the radical part the same.
By mastering coefficients, you will find simplifying complex expressions much more manageable.