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When we talk about simplifying radical expressions, we're usually referring to making them as simple as possible. A radical expression is any expression that includes a square root, cube root, etc. In this specific exercise, we are dealing with square roots.
Simplifying often involves reducing the expression to its simplest form where possible. While we may not be able to simplify \[ \sqrt{5} \] any further because 5 is a prime number and does not have a perfect square factor, the expression can still be simplified by combining coefficients of like radicals. Always look for opportunities to merge terms or simplify the radical part when possible.
Understanding how to handle these expressions becomes vital for ongoing algebra skills and prepares you for handling more complex operations later on.

• Always break down the expression to recognize what can and cannot be simplified.
• Use arithmetic to consolidate like terms, making the expression easier to handle.

Like terms are terms within an expression that have identical variables, and in the context of radicals, with identical radical parts. In this exercise, \( 8 \sqrt{5} \) and \( 11 \sqrt{5} \)are like terms because they share the same square root component, \( \sqrt{5} \).

• Like terms must have the same radicand (the number under the root). Here, \( \sqrt{5} \)
• They allow us to use simple addition or subtraction for simplification.

By focusing on the coefficients—numbers in front of the radical—you simplify the expression as you would with regular variables in algebra. For example,\( 8x + 11x = 19x \)is similar to \( 8 \sqrt{5} + 11 \sqrt{5} = 19 \sqrt{5} \).
Once identified, you manage the expression just as you would elementary arithmetic.

Adding radicals works similarly to adding any algebraic terms, but it's crucial they are like terms first. For radicals to be added together, they must have the same radicand. This condition simplifies things because it means you only need to add the coefficients, treating the radical part like a common unit. In our exercise, both terms have \( \sqrt{5} \), so we only need to focus on adding the coefficients 8 and 11:
\[ 8 \sqrt{5} + 11 \sqrt{5} = 19 \sqrt{5} \]
The process is straightforward:

• Ensure that the radicals share the same radicand.
• Add the coefficients while keeping the radical part constant.

Remember, if the radicals were different (say, \( \sqrt{3} \) and \( \sqrt{5} \)), you couldn't add them directly. Each different radicand would stay separate, much like you can't combine different variables without further information.