Understanding how to add and subtract radical expressions is essential in algebra, and it closely resembles the process of combining algebraic terms. When simplifying expressions with radicals, the key is to identify and combine like radicals, which share the same radicand (the number or expression inside the radical) and the same index (the root type, such as square root or cube root).

For instance, if you have an expression like \(2\sqrt{x} + 4\sqrt{x}\), you can combine these terms because they both contain the same radicand \(x\) and are both square roots. This results in \(6\sqrt{x}\). Similarly, constants can be combined in the same way, so \(3 + 8\) becomes \(11\).

However, caution is necessary when dealing with different types of radicals. For example, in the expression \(3\sqrt{7} + 2\sqrt{7} - \sqrt[3]{7}\), the first two terms can be combined to yield \(5\sqrt{7}\) since they share the same radicand and index. The term \(-\sqrt[3]{7}\) cannot be combined with the others because it has a different index (cube root versus square root), making it a different type of radical.

It’s also important to remember that you cannot merge separate radicals into one. For example, \(\sqrt{7} + \sqrt{7}\) does not equal \(\sqrt{14}\); instead, it simplifies to \(2\sqrt{7}\). This common mistake can lead to incorrect answers, so always ensure that you are only combining like terms.

In another example, consider \(9\sqrt[3]{x} + 4 - \sqrt{x}\). Here, \(9\sqrt[3]{x}\) and \(\sqrt{x}\) cannot be combined due to their different indexes, but the constant \(4\) can be combined with other constants if present. Thus, the expression simplifies to \(9\sqrt[3]{x} - \sqrt{x} + 4\).

In summary, when working with radical expressions, focus on identifying like radicals based on their radicands and indexes, combine them accordingly, and avoid merging separate radicals incorrectly. This approach will help you simplify radical expressions accurately.