In this guide
Squares, cubes and roots sit inside nearly every SSC CHSL maths question: Pythagoras, areas and volumes, simplification, compound interest and trigonometry. They are rarely the whole question, but they are often the slow step. Every square or cube you know by heart is a calculation you never do in the exam hall.
This guide has three parts: what to memorise, how to square and root numbers you haven't memorised, and how to handle surds.
What to memorise
Squares 1–30
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
Cubes 1–15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000, 1,331, 1,728, 2,197, 2,744, 3,375.
Also learn √2 ≈ 1.414, √3 ≈ 1.732 and √5 ≈ 2.236. They turn surd answers into decimals when the options are decimals.
Squaring quickly, and why each method works
Numbers ending in 5
For a number n5, the square is n × (n + 1) followed by 25. So 35² = 3 × 4 = 12, giving 1,225, and 85² = 72 followed by 25, giving 7,225.
Why: (10n + 5)² = 100n² + 100n + 25 = 100 × n(n + 1) + 25.
Numbers near 50 and 100
- (50 + a)² = 2,500 + 100a + a². So 52² = 2,500 + 200 + 4 = 2,704 and 48² = 2,500 − 200 + 4 = 2,304.
- (100 + a)² = 10,000 + 200a + a². So 103² = 10,609 and 98² = 10,000 − 400 + 4 = 9,604.
Why: these are (x + a)² = x² + 2ax + a² with x = 50 or 100, where x² and 2ax are easy.
The base method for any number
Use n² = (n + d)(n − d) + d², choosing d so that one factor is a round number. For 47², take d = 3: 50 × 44 + 9 = 2,200 + 9 = 2,209. For 63², take d = 3: 60 × 66 + 9 = 3,969.
Why: (n + d)(n − d) = n² − d², so adding d² back gives n².
Unit digits of squares
- A perfect square never ends in 2, 3, 7 or 8, and never in an odd number of zeros.
- Numbers ending in 1 or 9 have squares ending in 1. Numbers ending in 4 or 6 → 6. Numbers ending in 2 or 8 → 4. Numbers ending in 3 or 7 → 9. Numbers ending in 5 → 25.
- The digit sum of a perfect square reduces to 1, 4, 7 or 9. A number with digit sum 2, 3, 5, 6 or 8 is not a perfect square.
These rules let you reject "is this a perfect square?" options in seconds.
Square roots of perfect squares
- Use the unit digit to find the two possible last digits of the root.
- Use the size to fix the tens digit.
- Choose between the two candidates by comparing with the square of the number ending in 5 in that decade.
Cube roots of perfect cubes
The unit digit of a cube fixes the unit digit of its root: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0. Notice that 2 and 8 swap, and 3 and 7 swap; the rest stay the same. Then remove the last three digits and find the largest cube not exceeding what remains. That gives the tens digit.
Roots of decimals
Count decimal places. A square root halves them; a cube root divides them by three. So √0.0081 = 0.09 (four places become two), √1.44 = 1.2 and ∛0.027 = 0.3.
Surds
- Simplify by taking out square factors: √50 = √(25 × 2) = 5√2.
- Add only like surds: √12 + √27 = 2√3 + 3√3 = 5√3.
- Rationalise using (a + b)(a − b) = a² − b²: 1/(√3 − √2) = (√3 + √2)/(3 − 2) = √3 + √2.
- Compare sums of surds with the same total under the roots by squaring both. The pair whose numbers are closer together gives the larger sum.
Six worked questions
Q1. Find 65² and 105².
6 × 7 = 42, so 65² = 4,225. 10 × 11 = 110, so 105² = 11,025.
Q2. Find 97².
Near 100 with a = −3: 10,000 − 600 + 9 = 9,409.
Q3. Find √7,744.
80² = 6,400 and 90² = 8,100, so the root is in the 80s. Unit digit 4 means the root ends in 2 or 8. 85² = 7,225 is less than 7,744, so the root is above 85. Answer: 88.
Q4. Find ∛42,875.
Unit digit 5 → root ends in 5. Remove the last three digits to leave 42, which lies between 27 (3³) and 64 (4³). Tens digit 3. Answer: 35.
Q5. Simplify √0.0169 + √1.21.
0.13 + 1.1 = 1.23.
Q6. Which is larger: √10 + √3 or √8 + √5?
Square both. The first gives 13 + 2√30; the second gives 13 + 2√40. Since √40 > √30, √8 + √5 is larger.
A memorisation plan
| Days | Task |
|---|---|
| 1–3 | Squares 1–20, recited both ways (n → n² and n² → n) |
| 4–6 | Squares 21–30 and the √2, √3, √5 values |
| 7–9 | Cubes 1–15 and the cube-root unit-digit pairs |
| Every day after | Five-minute random self-test |
Practice
- Find 75².
- Find 47² using the base method.
- Find √5,329.
- Find ∛12,167.
- Find √0.0144.
- Simplify √75 + √48.
- Rationalise 1/(√3 − √2).
- Which of these is not a perfect square: 1,764; 2,116; 3,249; 4,357?
Answers:
- 5,625. 7 × 8 = 56, then 25.
- 2,209. 50 × 44 + 9.
- 73. Between 70² and 80²; ends in 3 or 7; 75² = 5,625 is too big, so 73.
- 23. Unit digit 7 → 3; 12 lies between 8 and 27 → 2.
- 0.12. Four decimal places become two.
- 9√3. 5√3 + 4√3.
- √3 + √2. Multiply top and bottom by √3 + √2; the denominator becomes 3 − 2 = 1.
- 4,357. A square can't end in 7. The others are 42², 46² and 57².
What to do next
- Write out squares to 30 and cubes to 15 from memory, and time yourself.
- Square ten random two-digit numbers a day using the base method.
- Practise ten square roots and ten cube roots until each takes under 15 seconds.
- Use these in simplification and mensuration, and revise the number system for divisibility rules.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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