91Ó°ÊÓ

In algebra, like terms are terms that have the exact same variables raised to the same power. For example, in the expression \( \textstyle \frac{1}{3} x^2 + 2x^2 - 5x\), \( \textstyle \frac{1}{3} x^2 \) and \( 2x^2 \) are like terms because they both contain \(x^2\). When dealing with radicals, like terms have the same radical part. For instance, in the expression \( \textstyle \frac{1}{2} \rho + 3 \rho - 2 \rho \), all terms are like terms because they all include \( \rho \). Identifying like terms is crucial when simplifying expressions, as it allows you to combine these terms to make the expression more straightforward. In our given problem, the terms \( \textstyle \frac{\rho \text{\textbackslash{5}}}{3\text{\textbackslash{7}}} \text{-{\textbackslash{4}}} \rho \text{\textbackslash{-{5}}} \text{7} \) need to be grouped by their radicals before any simplification can occur.

Radical operations involve performing mathematical operations of addition, subtraction, multiplication, and division with radical expressions. Just like with polynomials, we can combine like terms with radicals. Here are some basic rules to remember while performing radical operations:

• Only like terms with the same radicand (the number inside the radical) can be added or subtracted.
• Multiplication and division can be done with both like and unlike terms under certain conditions.

For the given expression \( \textstyle \rho \text{\textbackslash{5}} + 3 \rho \text{\textbackslash{7}} - 4 \rho \text{\textbackslash{5}} - 5 \rho \text{\textbackslash{7}} \), we first identify and group the like terms: \( \rho \text{\textbackslash{5}} \text{{and}} - 4 \rho \text{\textbackslash{5}} \) and \( 3 \rho \text{\textbackslash{7}} \text{{and}} - 5 \rho \text{\textbackslash{7}} \). This forms two groups that can be simplified separately.

Simplification is the process of reducing expressions to their simplest form. For radical expressions, this typically involves combining like terms and performing basic arithmetic operations. Here’s a step-by-step breakdown for the given expression: \( \textstyle \text{\textbackslash{5}} + 3 \rho \text{\textbackslash{7}} - \text{\textbackslash{4}} \rho \text{\textbackslash{5}} - 5 \rho \text{\textbackslash{7}} \)

• Step 1: Identify and group like terms: \( \rho \text{\textbackslash{5}} \text{{and}} - 4 \rho \text{\textbackslash{5}} \) and \( 3 \rho \text{\textbackslash{7}} \text{{and}} - 5 \rho \text{\textbackslash{7}} \).
• Step 2: Combine the like terms separately: \( (\rho \text{\textbackslash{5}} - 4 \rho \text{\textbackslash{5}}) \rightarrow \text{-} 3 \rho \text{\textbackslash{5}} \) and \( (3 \rho \text{\textbackslash{7}} - 5 \rho \text{\textbackslash{7}}) \rightarrow -2 \rho \text{\textbackslash{7}} \)
• Step 3: Combine the results: \(-3 \rho \text{\textbackslash{5}} \text{and} -2 \rho \text{\textbackslash{7}} \).

Everything combined together gives us the final simplified expression: \( \textstyle \text{-3} \rho \text{\textbackslash{5}} \text{-2} \rho \text{\textbackslash{7}} \). Always work through each step carefully to ensure accuracy in your solution.