To determine whether the result of each given mathematical expression is rational or irrational, let's analyze each expression step-by-step.
### Definitions:
1. Rational Number: A number that can be expressed as [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. Rational numbers include fractions, integers, and terminating or repeating decimals.
2. Irrational Number: A number that cannot be expressed as a simple fraction [tex]\(\frac{p}{q}\)[/tex]. These numbers have non-terminating, non-repeating decimal expansions. Examples include [tex]\(\sqrt{2}\)[/tex], [tex]\(\pi\)[/tex], etc.
### Expressions Analysis:
#### (a) [tex]\(34 + \sqrt{7}\)[/tex]
- [tex]\(34\)[/tex] is a rational number.
- [tex]\(\sqrt{7}\)[/tex] is an irrational number (since the square root of 7 is not a perfect square).
- The sum of a rational number and an irrational number is always irrational.
Result: Irrational
#### (b) [tex]\(\frac{12}{17} + \frac{4}{21}\)[/tex]
- [tex]\(\frac{12}{17}\)[/tex] and [tex]\(\frac{4}{21}\)[/tex] are both rational numbers.
- The sum of two rational numbers is always rational.
Result: Rational
#### (c) [tex]\(\sqrt{6} \times 23\)[/tex]
- [tex]\(\sqrt{6}\)[/tex] is an irrational number (since the square root of 6 is not a perfect square).
- [tex]\(23\)[/tex] is a rational number.
- The product of a nonzero rational number and an irrational number is always irrational.
Result: Irrational
#### (d) [tex]\(8 \times \frac{13}{19}\)[/tex]
- [tex]\(8\)[/tex] is a rational number.
- [tex]\(\frac{13}{19}\)[/tex] is a rational number.
- The product of two rational numbers is always rational.
Result: Rational
### Final Categorization in the Table:
\begin{tabular}{|c|c|c|}
\hline & \begin{tabular}{l}
Result is \\ Rational
\end{tabular} & \begin{tabular}{l}
Result is \\ Irrational
\end{tabular} \\
\hline (a) [tex]$34+\sqrt{7}$[/tex] & & [tex]\(\bigcirc\)[/tex] \\
\hline (b) [tex]$\frac{12}{17}+\frac{4}{21}$[/tex] & [tex]\(\bigcirc\)[/tex] & \\
\hline (c) [tex]$\sqrt{6} \times 23$[/tex] & & [tex]\(\bigcirc\)[/tex] \\
\hline (d) [tex]$8 \times \frac{13}{19}$[/tex] & [tex]\(\bigcirc\)[/tex] & \\
\hline
\end{tabular}
### Reasons:
- The sum of a rational and an irrational is always irrational.
- The sum of two rationals is always rational.
- The product of a nonzero rational and an irrational is always irrational.
- The product of two rationals is always rational.
