To determine which of the given numbers are irrational, let's analyze each number step-by-step:
1. Number: [tex]\(-4.21\)[/tex]
- Definition: A rational number can be expressed as a fraction [tex]\( \frac{a}{b} \)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
- Analysis: [tex]\(-4.21\)[/tex] can be written as [tex]\(-\frac{421}{100}\)[/tex], which is a fraction of two integers.
- Conclusion: [tex]\(-4.21\)[/tex] is a rational number.
2. Number: [tex]\(\sqrt{36}\)[/tex]
- Definition: An irrational number cannot be expressed as an exact fraction [tex]\( \frac{a}{b} \)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.
- Analysis: [tex]\(\sqrt{36}\)[/tex] equals [tex]\(6\)[/tex], which is an integer.
- Conclusion: [tex]\(\sqrt{36}\)[/tex] is a rational number.
3. Number: [tex]\(\sqrt{90}\)[/tex]
- Definition: An irrational number cannot be expressed as an exact fraction [tex]\( \frac{a}{b} \)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.
- Analysis: [tex]\(\sqrt{90}\)[/tex] cannot be simplified to an integer. It remains a non-repeating, non-terminating decimal.
- Conclusion: [tex]\(\sqrt{90}\)[/tex] is an irrational number.
4. Number: [tex]\(3 . \overline{8}\)[/tex] (repeating decimal)
- Definition: A repeating decimal number can be written as a fraction, hence it is rational.
- Analysis: [tex]\(3 . \overline{8}\)[/tex] can be written as [tex]\(\frac{38}{10}\)[/tex], then simplified to [tex]\(\frac{19}{5}\)[/tex].
- Conclusion: [tex]\(3 . \overline{8}\)[/tex] is a rational number.
5. Number: [tex]\(15\pi\)[/tex]
- Definition: An irrational number cannot be expressed as an exact fraction [tex]\( \frac{a}{b} \)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.
- Analysis: [tex]\(\pi\)[/tex] is known to be an irrational number. Multiplying an irrational number ([tex]\(\pi\)[/tex]) with a rational number ([tex]\(15\)[/tex]) results in an irrational number.
- Conclusion: [tex]\(15\pi\)[/tex] is an irrational number.
Based on the above analysis, the two irrational numbers are:
- [tex]\(\sqrt{90}\)[/tex]
- [tex]\(15\pi\)[/tex]
So, the final answer is that the two irrational numbers from the given set are [tex]\(\sqrt{90}\)[/tex] and [tex]\(15\pi\)[/tex].
