To determine whether each value [tex]\(a+b\)[/tex], [tex]\(a-b\)[/tex], and [tex]\(c^2\)[/tex] is always rational, never rational, or sometimes rational, we need to evaluate different scenarios involving the variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
### Value [tex]\(a + b\)[/tex]:
The sum of two numbers [tex]\(a + b\)[/tex] can be rational or irrational depending on the nature of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
- If [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are both rational, then [tex]\(a + b\)[/tex] is always rational.
- If [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are both irrational, [tex]\(a + b\)[/tex] can sometimes be rational (e.g., [tex]\( \sqrt{2} + (-\sqrt{2}) = 0 \)[/tex]) and sometimes irrational.
- If one of [tex]\(a\)[/tex] or [tex]\(b\)[/tex] is rational and the other is irrational, [tex]\(a + b\)[/tex] is always irrational.
Given the various possibilities, [tex]\(a + b\)[/tex] falls under the category of "Sometimes Rational."
### Value [tex]\(a - b\)[/tex]:
Similarly, the difference between two numbers [tex]\(a - b\)[/tex] can be rational or irrational based on [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- If [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are both rational, then [tex]\(a - b\)[/tex] is always rational.
- If [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are both irrational, [tex]\(a - b\)[/tex] can sometimes be rational (e.g., [tex]\( \sqrt{2} - \sqrt{2} = 0 \)[/tex]) and sometimes irrational.
- If one of [tex]\(a\)[/tex] or [tex]\(b\)[/tex] is rational and the other is irrational, [tex]\(a - b\)[/tex] is always irrational.
Given these cases, [tex]\(a - b\)[/tex] also falls under the category of "Sometimes Rational."
### Value [tex]\(c^2\)[/tex]:
The square of any real number [tex]\(c\)[/tex] will always be a real number. More specifically:
- If [tex]\(c\)[/tex] is a rational number, [tex]\(c^2\)[/tex] is a rational number.
- If [tex]\(c\)[/tex] is an irrational number, [tex]\(c^2\)[/tex] is still a rational number since squaring an irrational number does not necessarily yield an irrational number (e.g., [tex]\(\sqrt{2}^2 = 2\)[/tex]).
Thus, [tex]\(c^2\)[/tex] is always rational.
Based on these evaluations, the completed table would be:
\begin{tabular}{|c|c|c|c|}
\hline
Value & [tex]$a+b$[/tex] & [tex]$a-b$[/tex] & [tex]$c^2$[/tex] \\
\hline
Always Rational & & & X \\
\hline
Never Rational & & & \\
\hline
Sometimes Rational & X & X & \\
\hline
\end{tabular}
In this table, "X" denotes the appropriate cell to be selected for each value based on our analysis.
