Let's determine the nature of each expression's result step by step.
### (a) [tex]\(\sqrt{10} + 27\)[/tex]
- Reasoning: [tex]\(\sqrt{10}\)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers. Adding an irrational number to any rational number, in this case, 27, still results in an irrational number.
- Result: The result of [tex]\(\sqrt{10} + 27\)[/tex] is Irrational.
### (b) [tex]\(12 + \frac{4}{5}\)[/tex]
- Reasoning: Both 12 and [tex]\(\frac{4}{5}\)[/tex] are rational numbers. Rational numbers are defined as numbers that can be expressed as the quotient or fraction of two integers. The sum of two rational numbers remains rational.
- Result: The result of [tex]\(12 + \frac{4}{5}\)[/tex] is Rational.
### (c) [tex]\(21 \times \sqrt{3}\)[/tex]
- Reasoning: [tex]\(\sqrt{3}\)[/tex] is an irrational number. Multiplying any rational number, in this case, 21, with an irrational number results in an irrational number.
- Result: The result of [tex]\(21 \times \sqrt{3}\)[/tex] is Irrational.
### (d) [tex]\(\frac{12}{23} \times \frac{5}{11}\)[/tex]
- Reasoning: Both [tex]\(\frac{12}{23}\)[/tex] and [tex]\(\frac{5}{11}\)[/tex] are rational numbers. The product of two rational numbers is still rational.
- Result: The result of [tex]\(\frac{12}{23} \times \frac{5}{11}\)[/tex] is Rational.
Finally, summarizing the results in the given table format:
[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline & \text{Result is Rational} & \text{Result is Irrational} & \text{Reason} \\
\hline (a) \(\sqrt{10} + 27\) & & X & \begin{tabular}{c} Irrational \\ number added \\ to Rational \\ number gives \\ an Irrational number \end{tabular} \\
\hline (b) \(12 + \frac{4}{5}\) & X & & \begin{tabular}{c} Sum of \\ Rational \\ numbers \\ is Rational \end{tabular} \\
\hline (c) \(21 \times \sqrt{3}\) & & X & \begin{tabular}{c} Product \\ of Rational \\ and \\ Irrational \\ numbers \\ is Irrational \end{tabular} \\
\hline (d) \(\frac{12}{23} \times \frac{5}{11}\) & X & & \begin{tabular}{c} Product \\ of \\ Rational \\ numbers \\ is Rational \end{tabular} \\
\hline
\end{tabular}
\][/tex]
