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9 Arun says: a Is Arun correct? Give a reason for your answer. b Do you think sqrt(2(7)/(81)) is a rational number? Give a reason for your answer.

99 Arun says:\newlinea Is Arun correct? Give a reason for your answer.\newlineb Do you think 2781 \sqrt{2 \frac{7}{81}} is a rational number? Give a reason for your answer.

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Full solution

Q. 99 Arun says:\newlinea Is Arun correct? Give a reason for your answer.\newlineb Do you think 2781 \sqrt{2 \frac{7}{81}} is a rational number? Give a reason for your answer.
  1. Calculate Expression: To check if Arun is correct, we need to simplify the expression under the square root.\newlineCalculate 2(7)81\frac{2(7)}{81}.\newline2×7=142 \times 7 = 14.\newline1481=1481\frac{14}{81} = \frac{14}{81}.
  2. Simplify Square Root: Now, we simplify the square root: 1481\sqrt{\frac{14}{81}}. Since 8181 is a perfect square (9×99 \times 9), we can simplify the square root further. 1481=1481\sqrt{\frac{14}{81}} = \frac{\sqrt{14}}{\sqrt{81}}.
  3. Further Simplification: Simplify 81\sqrt{81} which is 99. So, 1481=149\sqrt{\frac{14}{81}} = \frac{\sqrt{14}}{9}.
  4. Final Evaluation: Since 14\sqrt{14} is not a perfect square, it cannot be simplified further.\newlineTherefore, 149\frac{\sqrt{14}}{9} is not a rational number because 14\sqrt{14} is an irrational number.
  5. Conclusion: Arun is incorrect; 2(7)81\sqrt{\frac{2(7)}{81}} is not a rational number.

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