Problem 174 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{11 c d}+5 \sqrt{11 c d}-9 \sqrt{11 c d} $$

Step-by-Step Solution

Answer

4 \sqrt{11 c d}

Step 1: Identify Like Terms

Notice that each term in the expression contains the same radical part \( \sqrt{11 c d} \). This common part allows combination via their coefficients.

Step 2: Combine Coefficients

Add and subtract the coefficients of the like terms: \( 8 + 5 - 9 \).

Step 3: Simplify Coefficients

Calculating \( 8 + 5 - 9 \) gives: \( 13 - 9 = 4 \).

Step 4: Combine with Radical Part

Combine the simplified coefficient with the common radical part: \( 4 \sqrt{11 c d} \).

Key Concepts

Like Terms

In algebra, 'like terms' are terms that have identical variables raised to the same power. For example, in the expression given, all terms have the same radical part, which is \( \sqrt{11 c d} \). Since these terms share the same radical, they are considered 'like terms'. This similarity allows them to be combined.
It's important to note that only like terms can be added or subtracted. If the radicals were different, they would not be combinable. Identifying like terms is a crucial first step in simplifying any algebraic expression.

Radical Expressions

A radical expression contains a root symbol, usually a square root\(\ (\sqrt{}) \), cube root \(\ ( \sqrt[3]{} \), and so on. In the given example, \( \sqrt{11 c d} \) is a square root of the product \( 11 c d \).
Radical expressions can often seem complicated, but they follow the same foundational rules as simpler algebraic expressions.

• Any constants or coefficients multiply directly with the radical.
• Like radical terms can be combined by adding or subtracting their coefficients.

Understanding how to manipulate these radical parts is key to simplifying expressions effectively.

Coefficients

Coefficients are the numerical factors in terms that contain variables or radicals. In our problem, 8, 5, and -9 are coefficients of the like terms \( 8 \sqrt{11 c d} \), \( 5 \sqrt{11 c d} \), and \( -9 \sqrt{11 c d} \).
When combining like terms, you focus on these coefficients. The process involves simple arithmetic operations like addition and subtraction. So, to solve our exercise:

• Identify the coefficients: 8, 5, -9.
• Perform the operations: \( 8 + 5 - 9 \).
• Combine the simplified coefficient with the radical part: 4 \( \sqrt{11 c d} \).

This approach makes the process systematic and easier to follow.

Combining Terms

Combining terms is the process of adding or subtracting like terms to simplify an algebraic expression. In this case:

• First, identify the like terms: \( 8 \sqrt{11 c d} \), \( 5 \sqrt{11 c d} \), and \( -9 \sqrt{11 c d} \).
• Then, focus on the coefficients: 8, 5, -9.
• Perform the arithmetic operation: \( 8 + 5 - 9 \).
• Combine the result with the radical: 4 \( \sqrt{11 c d} \).

This results in a simpler form, \( 4 \sqrt{11 c d} \). Remember, the operations are performed on the coefficients while keeping the common radical part unchanged. This method ensures accuracy and clarity in resolving algebraic expressions involving radicals.