Join the IDNLearn.com community and start finding the answers you need today. Our community provides accurate and timely answers to help you understand and solve any issue.
Sagot :
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].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.
