To categorize each of the given values as rational or irrational, let's break down each step-by-step:
1. Analyzing [tex]\(-\sqrt{36}\)[/tex]:
- The value [tex]\(\sqrt{36}\)[/tex] equals 6. Therefore, [tex]\(-\sqrt{36}\)[/tex] equals -6.
- Since -6 is an integer, it is also a rational number.
2. Analyzing [tex]\(\frac{9}{14}\)[/tex]:
- The fraction [tex]\(\frac{9}{14}\)[/tex] represents a ratio of two integers.
- Fractions where both the numerator and denominator are integers (and the denominator is not zero) are considered rational numbers.
3. Analyzing [tex]\(4.\overline{32}\)[/tex]:
- The notation [tex]\(4.\overline{32}\)[/tex] signifies a repeating decimal where 32 repeats indefinitely.
- Repeating decimals are representations of rational numbers.
4. Analyzing [tex]\(11\pi\)[/tex]:
- Pi ([tex]\(\pi\)[/tex]) is known to be an irrational number. Any non-zero multiple of an irrational number is also irrational.
- Therefore, [tex]\(11\pi\)[/tex] is irrational.
5. Analyzing [tex]\(\sqrt{14}\)[/tex]:
- The number 14 is not a perfect square, hence [tex]\(\sqrt{14}\)[/tex] is not an integer.
- The square root of any non-perfect square is an irrational number.
Now, filling in the table with these results:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline & \text{rational} & \text{irrational} \\
\hline
-\sqrt{36} & \text{1} & \text{0} \\
\hline
\frac{9}{14} & \text{1} & \text{0} \\
\hline
4 . \overline{32} & \text{1} & \text{0} \\
\hline
11\pi & \text{0} & \text{1} \\
\hline
\sqrt{14} & \text{0} & \text{1} \\
\hline
\end{tabular}
\][/tex]
The correct answer with the values filled in:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline & \text{rational} & \text{irrational} \\
\hline
-\sqrt{36} & 1 & 0 \\
\hline
\frac{9}{14} & 1 & 0 \\
\hline
4 . \overline{32} & 1 & 0 \\
\hline
11\pi & 0 & 1 \\
\hline
\sqrt{14} & 0 & 1 \\
\hline
\end{tabular}
\][/tex]
