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Understanding square roots is fundamental when simplifying radicals. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4.
When dealing with non-perfect squares like 8, it is often useful to break it down into smaller factors that include perfect squares. To simplify square roots, look for the largest perfect square factor. In the expression we're analyzing, \(\sqrt{8}\), which can be broken down as \(\sqrt{4 \times 2}\). Thus, it simplifies to \(2\sqrt{2}\), because \(\sqrt{4} = 2\).
Simplifying square roots makes combining like terms possible, allowing for easier computations later.

The concept of like terms is crucial when combining expressions. Like terms have the same variable raised to the same power, though they may have different coefficients. In expressions involving radicals, like terms are those that have the same radical part.
In our expression, after simplifying \(\sqrt{8}\) to \(2\sqrt{2}\), we end up with \(2\sqrt{2} + 3\sqrt{2}\). Here, both terms are considered like terms because they share the radical part \(\sqrt{2}\). This commonality allows us to combine them by simply adding the coefficients, much like traditional like terms such as\(2x + 3x = 5x\). Therefore, \(2\sqrt{2} + 3\sqrt{2}\) simplifies to \(5\sqrt{2}\).
Understanding how to identify and combine like terms with radicals is essential for simplifying radical expressions efficiently.

Mathematical expressions are combinations of numbers, variables, and operations. They do not have an equality or inequality sign, which distinguishes them from equations or inequalities. Their simplification often involves a series of steps depending on the elements within the expression.
In the given problem, the expression \(\sqrt{8} + 3\sqrt{2}\) involves both addition and square roots. By simplifying each square root into its simplest form and then combining like terms, the expression becomes more manageable.
The primary goal when working with mathematical expressions is to transform them into their simplest form while retaining equivalence. With practice, understanding how to simplify and manipulate these expressions becomes intuitive, leading to quicker and more accurate mathematical reasoning.