Square Roots & Cube Roots
Master important formulas and concepts with our comprehensive guide
Shortcut Techniques
Study MaterialShortcut Techniques – Square Roots & Cube Roots
Shortcut techniques for Square Roots and Cube Roots help candidates solve aptitude questions quickly and accurately in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.
Instead of lengthy calculations, these methods focus on:
- Fast square root calculations
- Quick cube root identification
- Prime factor shortcuts
- Perfect square and cube recognition
- Approximation techniques
- Algebraic identity methods
- Mental calculation tricks
Why Learn Shortcut Techniques?
- Improves calculation speed significantly.
- Reduces chances of calculation errors.
- Useful in simplification and approximation problems.
- Helps solve square and cube root questions mentally.
- Improves overall quantitative aptitude performance.
Shortcut #1: Memorize Important Squares
Learning squares of numbers up to 50 helps solve square root problems quickly.
| Number | Square |
|---|---|
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 15 | 225 |
| 20 | 400 |
| 25 | 625 |
| 30 | 900 |
✔ Memorizing squares improves simplification speed drastically.
Shortcut #2: Memorize Important Cubes
Learning cubes up to 20 helps in cube root identification.
| Number | Cube |
|---|---|
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 10 | 1000 |
Shortcut #3: Unit Digit Trick for Square Roots
The unit digit of a perfect square helps identify possible square roots quickly.
| Unit Digit of Square | Possible Unit Digit of Root |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
Example:
Find √729
Unit digit 9 means root ends with 3 or 7.
Since 25² = 625 and 30² = 900
729 lies between them.
Therefore:
√729 = 27
Shortcut #4: Unit Digit Trick for Cube Roots
Cube roots can be identified quickly using unit digits.
| Unit Digit of Cube | Unit Digit of Cube Root |
|---|---|
| 1 | 1 |
| 8 | 2 |
| 7 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 3 | 7 |
| 2 | 8 |
| 9 | 9 |
| 0 | 0 |
Example:
Find ∛2744
Last digit 4 means cube root ends in 4.
Ignoring last three digits:
2 lies between 1³ and 2³
Therefore:
∛2744 = 14
Shortcut #5: Prime Factorization Trick
For square roots:
- Make pairs of identical factors.
For cube roots:
- Make groups of three identical factors.
Example:
Find √144
144 = 2 × 2 × 2 × 2 × 3 × 3
Pair factors:
(2 × 2)(2 × 2)(3 × 3)
Answer:
2 × 2 × 3 = 12
Shortcut #6: Approximation Technique for Square Roots
Approximate square roots using nearby perfect squares.
Example:
√50
49 < 50 < 64
√49 = 7
√64 = 8
Therefore:
√50 ≈ 7.07
Shortcut #7: Product Property Trick
Break numbers into simpler factors.
Formula:
√(xy) = √x × √y
∛(xy) = ∛x × ∛y
Example:
√400
= √(100 × 4)
= 10 × 2
= 20
Shortcut #8: Decimal Root Trick
For square roots:
- Decimal places must appear in pairs.
For cube roots:
- Decimal places must appear in groups of three.
Example:
√0.25
= √25 / √100
= 5/10
= 0.5
Shortcut #9: Algebraic Identity Trick
Use identities for fast calculations involving roots.
| Identity | Formula |
|---|---|
| Square of Sum | (a + b)² = a² + b² + 2ab |
| Square of Difference | (a − b)² = a² + b² − 2ab |
| Difference of Squares | (a + b)(a − b) = a² − b² |
Example:
99 × 101
= (100 − 1)(100 + 1)
= 100² − 1²
= 9999
Shortcut #10: Recognizing Perfect Squares Quickly
A perfect square:
- Ends only with 0, 1, 4, 5, 6, or 9.
- Cannot end with 2, 3, 7, or 8.
- Has even number of trailing zeros.
✔ Useful for elimination techniques in MCQs.
Shortcut #11: Recognizing Perfect Cubes Quickly
A perfect cube:
- May end with any digit.
- Has trailing zeros in multiples of 3.
- Prime factor powers are multiples of 3.
Shortcut #12: Root Simplification Trick
Extract largest possible perfect square or cube from inside the root.
Example:
√72
= √(36 × 2)
= 6√2
Shortcut #13: Fast Square Calculation Near Base Numbers
When numbers are near 100:
Example:
103²
= (100 + 3)²
= 10000 + 600 + 9
= 10609
Shortcut #14: Fast Cube Calculation Near Base Numbers
Use:
(a + b)³ = a³ + b³ + 3ab(a + b)
Example:
11³
= (10 + 1)³
= 1000 + 1 + 3×10×1×11
= 1001 + 330
= 1331
Important Exam Tips
- Memorize squares up to 50 and cubes up to 20.
- Learn unit digit patterns carefully.
- Practice prime factorization regularly.
- Use approximation techniques for non-perfect roots.
- Apply algebraic identities for fast calculations.
- Recognize perfect squares and cubes quickly.
- Use elimination techniques in MCQ-based questions.
Shortcut techniques in Square Roots and Cube Roots help candidates improve calculation speed, reduce mistakes, and solve aptitude questions efficiently in competitive examinations.