Certainly, let's analyze each of the given expressions step-by-step to determine which one is a rational number.
### Expression 1: [tex]\( 3 \cdot \pi \)[/tex]
- [tex]\(\pi\)[/tex] (pi) is known to be an irrational number.
- Multiplying an irrational number by a rational number (such as 3) will result in an irrational number.
Hence, [tex]\( 3 \cdot \pi \)[/tex] is irrational.
### Expression 2: [tex]\( \frac{2}{3} + 9.26 \)[/tex]
- [tex]\(\frac{2}{3}\)[/tex] is a rational number because it can be expressed as the ratio of two integers.
- 9.26 is a terminating decimal, which means it can also be expressed as a ratio of two integers, making it a rational number.
The sum of two rational numbers is always rational.
Hence, [tex]\( \frac{2}{3} + 9.26 \)[/tex] is rational.
### Expression 3: [tex]\( \sqrt{45} + \sqrt{36} \)[/tex]
- [tex]\(\sqrt{45}\)[/tex] simplifies to [tex]\(3\sqrt{5}\)[/tex], which is an irrational number because [tex]\(\sqrt{5}\)[/tex] is irrational.
- [tex]\(\sqrt{36}\)[/tex] simplifies to 6, which is a rational number.
Adding an irrational number [tex]\(3\sqrt{5}\)[/tex] and a rational number 6 results in an irrational number.
Hence, [tex]\( \sqrt{45} + \sqrt{36} \)[/tex] is irrational.
### Expression 4: [tex]\( 14\overline{3} + 5.78765239\ldots \)[/tex]
- [tex]\( 14\overline{3} \)[/tex] is a repeating decimal, which is rational because it can be expressed as a fraction. Specifically, [tex]\( 14\overline{3} = 14.3333\ldots \)[/tex].
- [tex]\( 5.78765239 \ldots \)[/tex] is a terminating decimal, which is rational.
However, the analysis contains an inconsistency since it was previously concluded irrational, and this highlights a strict mathematical error if treated otherwise. Yet, the question clarity is circumscribed only onto which is strictly rational among provided terms.
Thus directly matched:
The rational number from the given expressions is:
[tex]\( \frac{2}{3} + 9.26 \)[/tex]
