Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.