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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}{|l|c|c|}
\hline
& \begin{tabular}{l}
Result is \\
Rational
\end{tabular} & \begin{tabular}{l}
Result is \\
Irrational
\end{tabular} \\
\hline
(a) [tex]$10+\sqrt{11}$[/tex] & & \\
\hline
(b) [tex]$-\sqrt{15}+\sqrt{7}$[/tex] & & \\
\hline
(c) [tex]$17 \times 11.65$[/tex] & & \\
\hline
(d) [tex]$-\sqrt{38} \times 2$[/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
Let's analyze each sum or product step-by-step to determine whether the result is rational or irrational, and then choose the appropriate reason for each.

### (a) [tex]\( 10 + \sqrt{11} \)[/tex]

1. Number Types:
- [tex]\( 10 \)[/tex] is a rational number.
- [tex]\( \sqrt{11} \)[/tex] is an irrational number because 11 is not a perfect square.

2. Sum of a Rational and an Irrational:
- The sum of a rational number and an irrational number is always irrational.

Result:
- The result is irrational.

### (b) [tex]\( -\sqrt{15} + \sqrt{7} \)[/tex]

1. Number Types:
- [tex]\( -\sqrt{15} \)[/tex] is an irrational number because 15 is not a perfect square.
- [tex]\( \sqrt{7} \)[/tex] is also an irrational number because 7 is not a perfect square.

2. Sum of Two Irrationals:
- The sum of two irrationals is sometimes rational, sometimes irrational. However, in this specific case, the sum [tex]\( -\sqrt{15} + \sqrt{7} \)[/tex] is irrational.

Result:
- The result is irrational.

### (c) [tex]\( 17 \times 11.65 \)[/tex]

1. Number Types:
- Both [tex]\( 17 \)[/tex] and [tex]\( 11.65 \)[/tex] are rational numbers.

2. Product of Two Rationals:
- The product of two rational numbers is always rational.

Result:
- The result is rational.

### (d) [tex]\( -\sqrt{38} \times 2 \)[/tex]

1. Number Types:
- [tex]\( -\sqrt{38} \)[/tex] is an irrational number because 38 is not a perfect square.
- [tex]\( 2 \)[/tex] is a rational number.

2. Product of a Nonzero Rational and an Irrational:
- The product of a nonzero rational number and an irrational number is always irrational.

Result:
- The result is irrational.

### Summary Table

[tex]\[ \begin{tabular}{|l|c|c|l|} \hline & \begin{tabular}{l} Result is \\ Rational \end{tabular} & \begin{tabular}{l} Result is \\ Irrational \end{tabular} & Reason \\ \hline (a) $10+\sqrt{11}$ & & \checkmark & The sum of a rational and an irrational is always irrational. \\ \hline (b) $-\sqrt{15}+\sqrt{7}$ & & \checkmark & The sum of two irrationals is sometimes rational, sometimes irrational. \\ \hline (c) $17 \times 11.65$ & \checkmark & & The product of two rationals is always rational. \\ \hline (d) $-\sqrt{38} \times 2$ & & \checkmark & The product of a nonzero rational and an irrational is always irrational. \\ \hline \end{tabular} \][/tex]

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