Question

Mia says that anytime there is a radical in an addition expression, it automatically has an irrational number as the sum. Is she correct

Answer 1

Mia's statement is not correct. While it's true that the sum of irrational numbers can sometimes be irrational, there are specific cases where the sum of a radical (irrational number) and a rational number can result in a rational number.

For example, consider the radical \(\sqrt{2}\), which is an irrational number. If we add it to another \(-\sqrt{2}\) (also an irrational number), the sum is:

\[
\sqrt{2} + (-\sqrt{2}) = 0
\]

In this case, the sum is a rational number (0).

Furthermore, if you take the expression \(\sqrt{2} + 1\) (where 1 is rational), the result \(\sqrt{2} + 1\) remains irrational.

So, the presence of a radical in an addition expression does not automatically mean the result will always be irrational; it depends on the specific numbers involved.