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 necessarily correct. The presence of a radical in an addition expression does not automatically mean that the sum is an irrational number.

For instance, consider the expression:

\[
\sqrt{4} + \sqrt{4} = 2 + 2 = 4
\]

In this case, \(\sqrt{4}\) is 2, which is a rational number, so the sum is also rational.

Another example is:

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

Here, the sum is 0, which is also a rational number.

However, if you take two irrational numbers that are not additive inverses, like \(\sqrt{2}\) and \(\sqrt{3}\):

\[
\sqrt{2} + \sqrt{3}
\]

This sum is indeed irrational.

In summary, while it's true that some sums involving radicals can be irrational, it is not a rule that any addition expression with radicals will always result in an irrational number. The specific values and operations involved must be examined to determine the nature of the sum.