Sure, let's identify whether each of the given numbers is rational or irrational.
(a) [tex]\(\sqrt{2}\)[/tex]:
The square root of 2 is an irrational number. It cannot be expressed as a fraction of two integers because its decimal representation is non-repeating and non-terminating.
(b) 7:
The number 7 is a rational number. It can be expressed as a fraction of two integers, [tex]\( \frac{7}{1} \)[/tex].
(c) 25:
The number 25 is a rational number. It can be expressed as a fraction of two integers, [tex]\( \frac{25}{1} \)[/tex].
(d) [tex]\(-\sqrt{18}\)[/tex]:
The square root of 18, simplified as [tex]\( \sqrt{9 \times 2} = 3\sqrt{2} \)[/tex], is an irrational number because [tex]\( \sqrt{2} \)[/tex] is irrational. Hence, [tex]\(-\sqrt{18}\)[/tex] is also irrational.
(e) 3:
The number 3 is a rational number. It can be expressed as a fraction of two integers, [tex]\( \frac{3}{1} \)[/tex].
(f) 0.285714:
This number appears to be a rational number because it is a terminating decimal. Terminating decimals can be expressed as fractions. Note: 0.285714 repeats after 6 digits, and its fractional representation is [tex]\( \frac{2}{7} \)[/tex].
(g) [tex]\(\frac{5}{8}\)[/tex]:
The fraction [tex]\(\frac{5}{8}\)[/tex] is a rational number because it is the quotient of two integers.
(h) 10:
The number 10 is a rational number. It can be expressed as a fraction of two integers, [tex]\( \frac{10}{1} \)[/tex].
(i) [tex]\(\frac{3}{3}\)[/tex]:
The fraction [tex]\(\frac{3}{3}\)[/tex] simplifies to 1, which is a rational number. It can be expressed as a fraction of two integers [tex]\( \frac{1}{1} \)[/tex].
Summarizing the results:
1. [tex]\(\sqrt{2}\)[/tex]: Irrational
2. 7: Rational
3. 25: Rational
4. [tex]\(-\sqrt{18}\)[/tex]: Irrational
5. 3: Rational
6. 0.285714: Irrational
7. [tex]\(\frac{5}{8}\)[/tex]: Irrational
8. 10: Rational
9. [tex]\(\frac{3}{3}\)[/tex]: Rational
