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Classifying Sums and Products as Rational or Irrational

For each sum or product, determine whether the result is a rational number or an irrational number. Then choose the appropriate reason for each.

\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] & & \\
\hline
(b) [tex]$\frac{12}{17}+\frac{4}{21}$[/tex] & & \\
\hline
(c) [tex]$\sqrt{6} \times 23$[/tex] & & \\
\hline
(d) [tex]$8 \times \frac{13}{19}$[/tex] & & \\
\hline
\end{tabular}

Choose one:
1. The sum of two rationals is always rational.
2. The sum of a rational and an irrational is always irrational.
3. The sum of two irrationals is sometimes rational, sometimes irrational.
4. The product of two rationals is always rational.
5. The product of a nonzero rational and an irrational is always irrational.
6. The product of two irrationals is sometimes rational, sometimes irrational.



Answer :

GinnyAnswer
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.

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