Certainly! Let's determine which of these expressions result in rational numbers by evaluating them step-by-step.
1. Expression: [tex]\(\sqrt{100} \cdot \sqrt{100}\)[/tex]
[tex]\[
\sqrt{100} = 10 \quad \text{and} \quad 10 \times 10 = 100
\][/tex]
The expression evaluates to 100, which is a rational number.
2. Expression: [tex]\(13.5 + \sqrt{81}\)[/tex]
[tex]\[
\sqrt{81} = 9 \quad \text{and} \quad 13.5 + 9 = 22.5
\][/tex]
The expression evaluates to 22.5, which is not a rational number but a decimal that can be expressed as a fraction [tex]\(\frac{45}{2}\)[/tex], thus rational.
3. Expression: [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]
[tex]\[
\sqrt{9} = 3 \quad \text{and} \quad \sqrt{729} = 27 \quad \text{so} \quad 3 + 27 = 30
\][/tex]
The expression evaluates to 30, which is a rational number.
4. Expression: [tex]\(\sqrt{64} + \sqrt{353}\)[/tex]
[tex]\[
\sqrt{64} = 8 \quad \text{and} \quad \sqrt{353} \text{ is irrational}
\][/tex]
An irrational number added to a rational number (8) remains irrational. Therefore, the expression is irrational.
5. Expression: [tex]\(\frac{1}{3} + \sqrt{216}\)[/tex]
[tex]\[
\sqrt{216} = 6 \sqrt{6} \quad (\text{irrational, would remain irrational even simplified})
\][/tex]
Adding an irrational number with [tex]\(\frac{1}{3}\)[/tex], a rational number, gives an irrational result.
6. Expression: [tex]\(\frac{3}{5} + 2.5\)[/tex]
[tex]\[
2.5 = \frac{5}{2} \quad \text{as a fraction}
\][/tex]
[tex]\[
\frac{3}{5} + \frac{5}{2} \quad (\text{Common denominator 10}) \quad = \frac{6}{10} + \frac{25}{10} = \frac{6 + 25}{10} = \frac{31}{10}
\][/tex]
The final result is a fraction, hence rational.
From our evaluations, the expressions that represent rational numbers are:
1. [tex]\(\sqrt{100} \cdot \sqrt{100}\)[/tex]
2. [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]
3. [tex]\(\frac{3}{5} + 2.5\)[/tex]
These correspond to the first, third, and sixth expressions on the list.
