Question

What must be true in order to combine radicals using addition or subtraction? Explain why it cannot always be immediately determined whether radicals can be combined with addition or subtraction.

Answer 1

In order to combine radicals using addition or subtraction, the radicals must have the same "radical part", which is similar to having like terms in regular algebra. Specifically, for two radicals to be combined, they must have the same index and the same radicand (the expression under the radical sign).

For example, the radicals \(\sqrt{2}\) and \(\sqrt{2}\) can be combined because they have the same radicand and index, resulting in:

\[
\sqrt{2} + \sqrt{2} = 2\sqrt{2}
\]

However, \(\sqrt{2}\) and \(\sqrt{3}\) cannot be combined because their radicands are different:

\[
\sqrt{2} + \sqrt{3} \text{ cannot be simplified further.}
\]

The reason it cannot always be immediately determined whether radicals can be combined lies in the complexity of the expressions involved. Radicals can be nested, simplified, or expressed in different forms. For instance, though \(\sqrt{8}\) and \(\sqrt{2}\) appear different at first glance, \(\sqrt{8}\) can be simplified to \(2\sqrt{2}\), which then makes it possible to combine:

\[
\sqrt{8} + \sqrt{2} = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2}.
\]

Therefore, to assess whether radicals can be combined, one often needs to simplify the radicals or rewrite them carefully to ensure that the radicands are the same. This process is not always straightforward, especially with more complicated expressions or when the radicals involve additional arithmetic operations.