Grade IX Number System Problems

Easy Level (35 Questions)

  1. Is 0 a rational number? Write it in p/q form.
  2. Write the next three natural numbers after 109.
  3. Classify: √4, √9, √16 as rational or irrational.
  4. Find five rational numbers between 0 and 1.
  5. Find three rational numbers between 2 and 3.
  6. Represent 5/2 on the number line.
  7. Without division, state whether 7/8 has terminating or non-terminating decimal.
  8. Write 0.75 as a fraction in lowest terms.
  9. Convert 0.4̅ (0.444…) to p/q form.
  10. Convert 0.12̅3 (0.123123…) to p/q.
  11. Simplify: 25^{1/2}.
  12. Simplify: 64^{1/3}.
  13. Simplify: 81^{3/4}.
  14. Simplify: (1/27)^{1/3}.
  15. Simplify: 8^{2/3}.
  16. Find √2 approximately on number line (up to 1 decimal).
  17. Is π rational or irrational? Give reason.
  18. Give one example each of terminating and non-terminating recurring decimal.
  19. Write 3 rational numbers between 1/4 and 1/3.
  20. Simplify: √36 + √49.
  21. Simplify: (√5)^2.
  22. Rationalize: 1/√2.
  23. Rationalize: 1/(√3 + 1).
  24. Simplify: (√3 + √2)^2.
  25. Find value: 16^{-3/4}.
  26. Express 0.001 as power of 10.
  27. Simplify: 2^5 × 2^{-3}.
  28. Simplify: (3^2)^3.
  29. Is √(4/9) rational? Justify.
  30. Locate √3 geometrically on number line.
  31. Convert 1.272727… to fraction.
  32. Simplify: 125^{2/3}.
  33. Find irrational number between 2 and 3.
  34. Classify 2 + √3 (rational/irrational).
  35. Simplify: 100^{1/2} × 100^{1/2}.

Medium Level (35 Questions)

  1. Prove that √3 is irrational (by contradiction).
  2. Prove that √5 is irrational.
  3. Find six rational numbers between 3/5 and 4/5.
  4. Express 0.235̅ in p/q form.
  5. Without actual division, classify decimal of 23/200.
  6. Rationalize denominator: 5/(√7 + √3).
  7. Rationalize: 1/(√5 + √2).
  8. Simplify: √(16/81) + √(25/100).
  9. If x = 2 + √3, find x + 1/x.
  10. Simplify: (√5 + √2)(√5 – √2).
  11. Simplify: (3 + √2)^2.
  12. Express 5.272727… as fraction.
  13. Simplify: 32^{2/5} × 32^{3/5}.
  14. Simplify: (729)^{1/6}.
  15. Rationalize: 1/(2 + √3 + √5).
  16. Locate √5 on number line (describe construction).
  17. Prove: √2 + √3 is irrational.
  18. Find three different irrationals between 0.12 and 0.13.
  19. Simplify: (64/125)^{-1/3}.
  20. If a = 3 + 2√2, find a^2 + 1/a^2.
  21. Rationalize: (√5 + √3)/(√5 – √3).
  22. Simplify: √(50) – √(18) + √(32).
  23. Convert 0.47̅ to p/q.
  24. Simplify: 243^{2/5}.
  25. Find value: (1/8)^{-2/3}.
  26. Rationalize denominator of 1/(√7 – √6).
  27. Simplify: (√3 + √5)^2 – (√3 – √5)^2.
  28. Express 1.363636… as fraction.
  29. Simplify: 10^{3/2} × 10^{-1/2}.
  30. Find rational number between √2 and √3.
  31. Rationalize: 4/(3 + 2√2).
  32. Simplify: (125 × 64)^{1/3}.
  33. If x = 1/(2 + √3), rationalize and find value.
  34. Prove that 5 – √7 is irrational.
  35. Simplify: √(72) + √(50) – √(8).

Difficult Level (30 Questions)

  1. Rationalize denominator: 1/(√3 + √5 + √7).
  2. If x = 3 + √8, find x^2 + 1/x^2 + x + 1/x.
  3. Simplify: 1/(√4 + √3) + 1/(√4 – √3).
  4. Rationalize: 5 + 2√3 / (7 + 4√3).
  5. Express 0.235235235… (bar on 235) as p/q.
  6. Prove that √2 + √5 is irrational.
  7. Simplify: (√3 + √2 + √5)(√3 + √2 – √5).
  8. Rationalize denominator of 1/(2√3 + 3√2 + √5).
  9. If a = √3 + √2, find a^4 – 10a^2 + 1.
  10. Simplify: (81/16)^{-3/4} × (25/9)^{3/2}.
  11. Find value of x if 3^{x–1} × 9^{x+1} = 27^{2x–1}.
  12. Rationalize: 1/(√10 + √6 + √15 + √9).
  13. Simplify and express in simplest form: √(18 + 10√3).
  14. If x = 1 + √2 + √3, find minimal polynomial or simplified power.
  15. Prove √7 is irrational and then show 2 + 3√7 irrational.
  16. Rationalize: (√5 – √3 – √2)/(√5 + √3 + √2).
  17. Simplify: (√7 + √5 + √3 + √2)^2 – (√7 + √5 – √3 – √2)^2.
  18. Express 0.123456789101112… (Champertnowne like, but simple repeating block) wait – better: 0.123̅456̅ as fraction (two bars).
  19. Find x if (√3)^x = 27^{1/3} × 9^{-1/2}.
  20. Rationalize denominator: 1/(4 + √15 + √10 + √6).
  21. Simplify: √(7 + 4√3).
  22. If a + b = √5 + √3, a – b = √5 – √3, find ab.
  23. Prove that (√2 + √3)^6 + (√2 – √3)^6 is integer.
  24. Rationalize multi-step: 1/(√8 + √6 + √3 + √2).
  25. Simplify: (32)^{3/5} + (243)^{2/5} – (1/32)^{-3/5}.
  26. Find rationalised form and value: 1/(3 + 2√2 + √3).
  27. Prove 3 + 2√5 is irrational using contradiction.
  28. Simplify √(12 + 6√3 + 4√2 + 2√6).
  29. If x = √(a + √b) + √(a – √b), find x^2.
  30. Combined: Rationalize 1/(√7 – √5) + 1/(√5 – √3) + 1/(√3 – √7) and simplify.