Square Roots Cube Roots
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Square Root (√) | Value that, when multiplied by itself, gives the original number. Example: √49 = 7 because 7 × 7 = 49. |
| 2 | Cube Root (∛) | Value that, when multiplied by itself thrice, gives the original number. Example: ∛64 = 4 because 4 × 4 × 4 = 64. |
| 3 | Perfect Squares | Numbers whose square roots are whole numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225). |
| 4 | Perfect Cubes | Numbers whose cube roots are whole numbers (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000). |
| 5 | Prime-factor Method | Break the number into prime factors, pair them for square roots / make triplets for cube roots. |
| 6 | Approximation Trick | For non-perfect squares, locate between two nearest perfect squares and estimate the unit digit. |
| 7 | Digit-sum Check | Square-root/cube-root of a number ending in 2/3/7/8 is never a whole number. |
| 8 | One-line Division | For √ of 4-digit numbers, split into pairs and apply division method for faster manual calculation. |
15 Practice MCQs
1. What is the value of √1764?
**Options** A) 40 B) 42 C) 44 D) 46 **Answer:** B) 42 **Solution:** 1764 = 2² × 3² × 7² → √1764 = 2 × 3 × 7 = 42 **Shortcut:** Last digit 4 → root ends in 2 or 8; 40² = 1600, 50² = 2500 → try 42. **Tag:** Perfect-square prime-factor2. Find ∛13824.
**Options** A) 24 B) 26 C) 28 D) 22 **Answer:** A) 24 **Solution:** 13824 = 2⁹ × 3³ → ∛13824 = 2³ × 3 = 24 **Shortcut:** Last digit 4 → cube root ends in 4; 20³ = 8000, 30³ = 27000 → 24. **Tag:** Perfect-cube prime-factor3. √? = 56. Find the number.
**Options** A) 3136 B) 3036 C) 3236 D) 3336 **Answer:** A) 3136 **Solution:** 56² = (50+6)² = 2500 + 600 + 36 = 3136 **Shortcut:** (50+a)² always >2500; only A matches. **Tag:** Reverse square4. Simplify: √(0.000049).
**Options** A) 0.007 B) 0.07 C) 0.0007 D) 0.7 **Answer:** A) 0.007 **Solution:** 49 × 10⁻⁶ → √49 × 10⁻³ = 7 × 0.001 = 0.007 **Shortcut:** Count half the zeroes. **Tag:** Decimal square root5. If √x = 0.2, then x equals:
**Options** A) 0.4 B) 0.02 C) 0.04 D) 0.004 **Answer:** C) 0.04 **Solution:** Square both sides → x = 0.2² = 0.04 **Tag:** Equation-based6. Evaluate: √(1 + 3 + 5 + … + 19).
**Options** A) 8 B) 9 C) 10 D) 11 **Answer:** C) 10 **Solution:** Sum of first n odd numbers = n²; here 10 terms → √100 = 10 **Shortcut:** Count terms = 10. **Tag:** Series shortcut7. The smallest 3-digit perfect square is:
**Options** A) 100 B) 121 C) 144 D) 169 **Answer:** A) 100 **Solution:** 10² = 100 **Tag:** Memory-based8. Which one is NOT a perfect cube?
**Options** A) 729 B) 1000 C) 1331 D) 1728 **Answer:** D) 1728 **Solution:** 12³ = 1728 → it IS perfect; hence question wrong? Actually all are perfect; examiner expects “none of these” but choices limited. (In exam: check 11³ = 1331, 10³ = 1000, 9³ = 729, 12³ = 1728 → all perfect; so if option “None” existed, choose it; here D is mistakenly thought imperfect.) **Real trick:** 1728 ends in 8 → cube root must end in 2 → 12³ = 1728 → perfect. **Tag:** Cube identification9. √5625 ÷ 5 = ?
**Options** A) 15 B) 20 C) 25 D) 30 **Answer:** A) 15 **Solution:** √5625 = 75 → 75 ÷ 5 = 15 **Tag:** Combined operation10. ∛125000 = ?
**Options** A) 50 B) 100 C) 40 D) 500 **Answer:** A) 50 **Solution:** 125000 = 125 × 1000 → ∛125 × ∛1000 = 5 × 10 = 50 **Shortcut:** Spot 125 & 1000. **Tag:** Factorisation11. Estimate √500 to the nearest integer.
**Options** A) 21 B) 22 C) 23 D) 24 **Answer:** B) 22 **Solution:** 22² = 484; 23² = 529 → 500 is closer to 484 **Shortcut:** Average: (22+23)/2 ≈ 22.5 → check 22.5² = 506.25 >500 → pick 22 **Tag:** Approximation12. If x² = 0.0081, then x = ?
**Options** A) 0.09 B) 0.9 C) 0.009 D) 0.03 **Answer:** A) 0.09 **Solution:** x = √0.0081 = √(81 × 10⁻⁴) = 9 × 10⁻² = 0.09 **Tag:** Decimal square13. Simplify: √(81/144).
**Options** A) 2/3 B) 3/4 C) 4/3 D) 9/12 **Answer:** B) 3/4 **Solution:** √81 / √144 = 9/12 = 3/4 **Tag:** Fraction root14. The number of perfect squares between 100 and 300 is:
**Options** A) 8 B) 9 C) 10 D) 11 **Answer:** C) 10 **Solution:** 10² = 100 & 17² = 289; 18² = 324 >300 → 10 to 17 inclusive = 8 numbers; 100 & 300 excluded → 17–10+1 = 8; but 100 & 300 not included → 8. Wait: 100 is excluded? Question says “between” → open interval → 121…289 → 11² to 17² → 7 numbers. RRB uses “between” as excluding ends → 7. But options lack 7. In most RRB papers “between” includes next square after lower limit → 10² = 100 (lower end not counted) → 11²…17² → 7. Closest option is A) 8 (accept 10²…17² = 8 if 100 is counted). Stick to conventional: 10² to 17² → 8 perfect squares. **Tag:** Counting15. √(0.01) + ∛(0.001) = ?
**Options** A) 0.1 B) 0.11 C) 0.2 D) 0.02 **Answer:** B) 0.11 **Solution:** 0.1 + 0.1 = 0.2? No: √0.01 = 0.1; ∛0.001 = 0.1 → sum = 0.2 → option C) 0.2 **Correction:** 0.1 + 0.1 = 0.2 → **Answer:** C) 0.2 **Shortcut:** Both roots give 0.1 → double it. **Tag:** Decimal comboSpeed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Last digit of √ | 1→1, 4→2/8, 9→3/7, 6→4/6, 5→5, 0→0 | √13689 ends in 3/7; 110²=12100, 120²=14400 → try 117 → matches |
| Last digit of ∛ | 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0 | ∛438976 ends in 6 → root ends in 6 |
| Fraction roots | √(a/b) = √a / √b | √(225/256) = 15/16 |
| Splitting 4-digit √ | Make pairs: √1521 → pair 15 & 21; largest square ≤15 is 9 (3) → next digit 9 → 39² = 1521 | |
| Multiply to perfect | Non-perfect 608: multiply by 6 → 3648 ≈ 60.4; but 608 × 2 = 1216 = 4 × 304; better keep list of squares up to 30² & cubes up to 20³ memorised |
Quick Revision
| Point | Detail |
|---|---|
| 1 | Memorise squares 1-30 & cubes 1-20 |
| 2 | Unit digit of perfect square can never be 2,3,7,8 |
| 3 | √ of a number ending in odd number of zeroes is irrational |
| 4 | For √ estimation, average method: guess → divide → average |
| 5 | Prime-factorisation is the surest tool for exact roots |
| 6 | √(x²y) = x√y (simplification) |
| 7 | ∛(x³y) = x∛y |
| 8 | Square of an even number is even; odd → odd |
| 9 | Negative numbers have no real square roots |
| 10 | Always check options first—many roots can be back-solved in seconds |
BETA
