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When working with algebraic expressions, simplifying radicals is a fundamental skill that helps clear up the complexity of an equation. To simplify a radical, we aim to express it in simplest radical form where the radicand, the number under the radical sign, has no perfect square factors.

For example, the radical \begin{small}\( \sqrt{50} \)\end{small}can be simplified by recognizing that 50 can be expressed as \begin{small}\( 25 * 2 \)\end{small}, and since 25 is a perfect square, we have \begin{small}\( \sqrt{50} = \sqrt{25} * \sqrt{2} = 5\sqrt{2} \)\end{small}. In general, the process involves finding the largest perfect square factor, taking its square root outside the radical, and multiplying it by the remaining radical. Always look for the greatest square factors when simplifying, as this will result in the simplest form.

Square roots are the opposite of squaring a number. When we square a number, we multiply it by itself; when we take the square root, we look for a number that, when multiplied by itself, gives the original number. The square root of a number \begin{small}\( x \)\end{small} is denoted as \begin{small}\( \sqrt{x} \)\end{small}. For positive real numbers, the square root has two values: one positive and one negative, since both \begin{small}\( (\sqrt{x})^2 \)\end{small} and \begin{small}\( (-\sqrt{x})^2 \)\end{small} will equal \begin{small}\( x \)\end{small}.

However, in most algebraic expressions, we use the principal square root, which is the positive root. Square roots become even more relevant when combined with variables, as in the exercise \begin{small}\( \sqrt{x y} \)\end{small}, where \begin{small}\( x \)\end{small} and \begin{small}\( y \)\end{small} are variables.

Identifying and combining like terms is a critical step in simplifying algebraic expressions. Like terms have the same variable raised to the same power, even if they have different coefficients. To simplify an expression by combining like terms, simply add or subtract the coefficients, while keeping the variable part unchanged.

In the given example, the sequence of steps shows that the terms \begin{small}\( 3\sqrt{x y} \)\end{small} and \begin{small}\( -8\sqrt{x y} \)\end{small} are like terms. Even though they have different coefficients, the variable part \begin{small}\( \sqrt{x y} \)\end{small} is the same. By subtracting the coefficients, we end up with \begin{small}\( -5\sqrt{x y} \)\end{small}. It's important to pay attention to signs and keep track of plus and minus through combining like terms to ensure accuracy in simplification.