When working with radicals, it's essential to understand how to add and subtract them, especially when they are not like radicals. Like radicals have the same radicand (the number inside the square root) and index, allowing for straightforward addition or subtraction. For instance, combining \(3\sqrt{5}\) and \(4\sqrt{5}\) results in \(7\sqrt{5}\) because the radicands are the same.

However, when faced with unlike radicals, such as \(\sqrt{5}\) and \(\sqrt{20}\), the first step is to simplify them. Simplification involves breaking down the radicals into their prime factors to identify any perfect squares. In the case of \(\sqrt{20}\), it can be expressed as \(\sqrt{4 \cdot 5}\), which simplifies to \(2\sqrt{5}\). This transformation allows us to rewrite the original expression as \(\sqrt{5} + 2\sqrt{5}\), which can then be combined to yield \(3\sqrt{5}\).

Another example involves \(5\sqrt{2}\) and \(\sqrt{18}\). Here, \(\sqrt{18}\) can be simplified to \(\sqrt{9 \cdot 2}\), resulting in \(3\sqrt{2}\). Thus, the expression becomes \(5\sqrt{2} - 3\sqrt{2}\), which simplifies to \(2\sqrt{2}\).

In a more complex scenario with \(\sqrt{18}\) and \(\sqrt{50}\), both radicals need simplification. \(\sqrt{18}\) simplifies to \(3\sqrt{2}\) and \(\sqrt{50}\) simplifies to \(5\sqrt{2}\). This results in the expression \(3\sqrt{2} + 5\sqrt{2}\), which combines to \(8\sqrt{2}\).

In summary, the key to adding or subtracting unlike radicals is to simplify each radical first, transforming them into like radicals before performing the operation. This method not only clarifies the process but also ensures accuracy in calculations.