Question
Which of the following numbers is irrational?
(a) \( \sqrt{\frac{4}{9}} \)
(b) \( \frac{\sqrt{1250}}{\sqrt{8}} \)
(c) \( \sqrt{8} \)
(d) \( \frac{\sqrt{24}}{\sqrt{6}} \)
Solution
Option (a): \( \sqrt{\frac{4}{9}} \)
\[ \sqrt{\frac{4}{9}} = \frac{2}{3} \]
✔ Rational
Option (b): \( \frac{\sqrt{1250}}{\sqrt{8}} \)
\[ = \sqrt{\frac{1250}{8}} = \sqrt{156.25} = 12.5 \]
✔ Rational
Option (c): \( \sqrt{8} \)
\[ \sqrt{8} = 2\sqrt{2} \]
✔ Irrational
Option (d): \( \frac{\sqrt{24}}{\sqrt{6}} \)
\[ = \sqrt{\frac{24}{6}} = \sqrt{4} = 2 \]
✔ Rational
Final Answer
✔ Correct option: (c) \( \sqrt{8} \)