91Ó°ÊÓ

In algebra, combining like terms is a critical skill. This process involves grouping terms that have the same variables raised to the same power, allowing you to simplify expressions. In our original exercise, we observe expressions like \(3 \sqrt{5}\) and \(5 \sqrt{5}\) as `like terms` because both include the square root of 5. Similarly, terms such as \(-4 \sqrt{11}\) and \(12 \sqrt{11}\) are grouped together.

Here's how you can do it effectively:

• Identify like terms: Find terms with identical variables or expressions, such as \(\sqrt{5}\).
• Group them: Write down these terms together so you can work on them more easily.


• Add or subtract as needed: Once grouped, combine by performing operations on the coefficients.

The goal is to end up with fewer terms that better encapsulate the variables or expressions involved.

Coefficients are the numerical factors that multiply the variable or expression in a term. In the expression \(3 \sqrt{5}\), the number 3 is the coefficient of \(\sqrt{5}\). Similarly, in \(-4 \sqrt{11}\), -4 is the coefficient of \(\sqrt{11}\).
Understanding coefficients can greatly simplify solving algebraic equations, helping you to deal with terms more efficiently.

The exercise showcases how easy it is to work with coefficients:

• Identify: Look at each term and identify the number in front of the variable or expression.
• Combine: Add or subtract these numbers as needed to simplify.

In our exercise, when we grouped the terms \(3 \sqrt{5}\) and \(5 \sqrt{5}\), we added their coefficients, 3 and 5, leading to a new coefficient of 8 for \(\sqrt{5}\). Similarly, for \(-4 \sqrt{11}\) and \(12 \sqrt{11}\), the coefficients -4 and 12 combined to yield 8 for \(\sqrt{11}\). By mastering coefficients, your algebraic manipulation skills will greatly improve.

Square roots are fundamental in mathematics and frequently appear in algebraic expressions. The expression \( \sqrt{n} \) signifies a number which, when multiplied by itself, results in \( n \). In our problem, terms like \( \sqrt{5} \) and \( \sqrt{11} \) cannot be simplified further since 5 and 11 are prime numbers and don't have natural square roots.

This topic involves essential insights that can guide you when dealing with square roots:

• Simplification: Always check if the number under the square root (known as the radicand) can be simplified. For example, \( \sqrt{8} \) simplifies to \(2 \sqrt{2}\).


• Operations: Treat square roots like variables when combining like terms; ensure that only square roots with the same radicand are combined.

In the exercise, since the square roots involve different numbers, \(5\) and \(11\), we only grouped like terms sharing the same radicand \( \sqrt{5} \) or \( \sqrt{11} \). This respect for square roots' properties helps achieve a simplified yet accurate expression.