Answer :
Let's analyze each of the options to determine whether they are rational or irrational.
### Option 1: [tex]\(3 \cdot \pi\)[/tex]
- [tex]\(\pi\)[/tex] (pi) is known to be an irrational number. An irrational number cannot be expressed as a fraction of two integers.
- Multiplying [tex]\( \pi \)[/tex] by an integer (in this case, 3) does not change its irrationality.
- Hence, [tex]\(3 \cdot \pi\)[/tex] is also an irrational number.
Conclusion: [tex]\(3 \cdot \pi\)[/tex] is not a rational number.
### Option 2: [tex]\(\frac{2}{3} + 9.26\)[/tex]
- [tex]\(\frac{2}{3}\)[/tex] is a rational number because it is a fraction of two integers.
- 9.26 is a terminating decimal, which is a form of a rational number because it can also be expressed as a fraction (in this case, [tex]\(\frac{926}{100}\)[/tex]).
- Adding two rational numbers together results in another rational number.
Conclusion: [tex]\(\frac{2}{3} + 9.26\)[/tex] is a rational number.
### Option 3: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex]
- [tex]\(\sqrt{45}\)[/tex] can be simplified to [tex]\(3\sqrt{5}\)[/tex]. Since [tex]\(\sqrt{5}\)[/tex] is an irrational number, [tex]\(3\sqrt{5}\)[/tex] is also irrational.
- [tex]\(\sqrt{36}\)[/tex] simplifies to 6, which is a rational number.
- The sum of an irrational number ([tex]\(3\sqrt{5}\)[/tex]) and a rational number (6) is always irrational.
Conclusion: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex] is not a rational number.
### Option 4: [tex]\(14.\overline{3} + 5.78765239\)[/tex]
- [tex]\(14.\overline{3}\)[/tex] (14.333...) is a repeating decimal, which is a rational number. Repeating decimals can be expressed as fractions.
- 5.78765239 is a terminating decimal, another form of a rational number.
- Adding two rational numbers together results in another rational number.
Conclusion: [tex]\(14.\overline{3} + 5.78765239\)[/tex] is a rational number.
### Final Summary
- Option 1: [tex]\(3 \cdot \pi\)[/tex] is not rational.
- Option 2: [tex]\(\frac{2}{3} + 9.26\)[/tex] is rational.
- Option 3: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex] is not rational.
- Option 4: [tex]\(14.\overline{3} + 5.78765239\)[/tex] is rational.
Therefore, the rational expressions among the given options are [tex]\(\frac{2}{3} + 9.26\)[/tex] and [tex]\(14.\overline{3} + 5.78765239\)[/tex].
### Option 1: [tex]\(3 \cdot \pi\)[/tex]
- [tex]\(\pi\)[/tex] (pi) is known to be an irrational number. An irrational number cannot be expressed as a fraction of two integers.
- Multiplying [tex]\( \pi \)[/tex] by an integer (in this case, 3) does not change its irrationality.
- Hence, [tex]\(3 \cdot \pi\)[/tex] is also an irrational number.
Conclusion: [tex]\(3 \cdot \pi\)[/tex] is not a rational number.
### Option 2: [tex]\(\frac{2}{3} + 9.26\)[/tex]
- [tex]\(\frac{2}{3}\)[/tex] is a rational number because it is a fraction of two integers.
- 9.26 is a terminating decimal, which is a form of a rational number because it can also be expressed as a fraction (in this case, [tex]\(\frac{926}{100}\)[/tex]).
- Adding two rational numbers together results in another rational number.
Conclusion: [tex]\(\frac{2}{3} + 9.26\)[/tex] is a rational number.
### Option 3: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex]
- [tex]\(\sqrt{45}\)[/tex] can be simplified to [tex]\(3\sqrt{5}\)[/tex]. Since [tex]\(\sqrt{5}\)[/tex] is an irrational number, [tex]\(3\sqrt{5}\)[/tex] is also irrational.
- [tex]\(\sqrt{36}\)[/tex] simplifies to 6, which is a rational number.
- The sum of an irrational number ([tex]\(3\sqrt{5}\)[/tex]) and a rational number (6) is always irrational.
Conclusion: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex] is not a rational number.
### Option 4: [tex]\(14.\overline{3} + 5.78765239\)[/tex]
- [tex]\(14.\overline{3}\)[/tex] (14.333...) is a repeating decimal, which is a rational number. Repeating decimals can be expressed as fractions.
- 5.78765239 is a terminating decimal, another form of a rational number.
- Adding two rational numbers together results in another rational number.
Conclusion: [tex]\(14.\overline{3} + 5.78765239\)[/tex] is a rational number.
### Final Summary
- Option 1: [tex]\(3 \cdot \pi\)[/tex] is not rational.
- Option 2: [tex]\(\frac{2}{3} + 9.26\)[/tex] is rational.
- Option 3: [tex]\(\sqrt{45} + \sqrt{36}\)[/tex] is not rational.
- Option 4: [tex]\(14.\overline{3} + 5.78765239\)[/tex] is rational.
Therefore, the rational expressions among the given options are [tex]\(\frac{2}{3} + 9.26\)[/tex] and [tex]\(14.\overline{3} + 5.78765239\)[/tex].
