Sure, let's evaluate each expression to determine whether the result is a rational or an irrational number, and then choose the appropriate reasons for each.
### (a) [tex]\(\frac{2}{11} + \frac{17}{19}\)[/tex]
- Result: Rational
- Reason: The sum of two rational numbers is always rational.
Since [tex]\(\frac{2}{11}\)[/tex] and [tex]\(\frac{17}{19}\)[/tex] are both rational numbers (fractions of integers), their sum is also a rational number.
### (b) [tex]\(16 + \sqrt{13}\)[/tex]
- Result: Irrational
- Reason: The sum of a rational number and an irrational number is always irrational.
Here, [tex]\(16\)[/tex] is a rational number, and [tex]\(\sqrt{13}\)[/tex] is an irrational number, thus their sum is irrational.
### (c) [tex]\(20 \times \frac{8}{9}\)[/tex]
- Result: Rational
- Reason: The product of two rational numbers is always rational.
Since [tex]\(20\)[/tex] and [tex]\(\frac{8}{9}\)[/tex] are both rational numbers, their product is also a rational number.
### (d) [tex]\(\sqrt{2} \times 38\)[/tex]
- Result: Irrational
- Reason: The product of a rational number and an irrational number is always irrational.
Here, [tex]\(\sqrt{2}\)[/tex] is an irrational number, and [tex]\(38\)[/tex] is a rational number, so their product is irrational.
The completed table is as follows:
\begin{tabular}{|l|l|l|l|}
\hline & \begin{tabular}{l}
Result is \\
Rational
\end{tabular} & \begin{tabular}{l}
Result is \\
Irrational
\end{tabular} & \\
\hline (a) [tex]$\frac{2}{11}+\frac{17}{19}$[/tex] & X & & Reason 1: The sum of two rational numbers is rational\\
\hline (b) [tex]$16+\sqrt{13}$[/tex] & & X & Reason 2: The sum of a rational number and an irrational number is irrational \\
\hline (c) [tex]$20 \times \frac{8}{9}$[/tex] & X & & Reason 3: The product of two rational numbers is rational \\
\hline (d) [tex]$\sqrt{2} \times 38$[/tex] & & X & Reason 4: The product of a rational number and an irrational number is irrational \\
\hline
\end{tabular}
