Why it matters
What is 8 + 2 × 5? One student says 50, another says 18. The answer is 18. Mathematics has a fixed order of operations so that everyone gets the same answer from the same expression. That order is called BODMAS.
- Simplification means turning a long expression with many numbers and signs into one single number, step by step, in the correct order.
- A square of a number is the number multiplied by itself: 7² = 7 × 7 = 49.
- A square root is the reverse: √49 = 7, because 7 × 7 = 49.
- A cube is a number used three times in multiplication (4³ = 64), and a cube root is its reverse (∛64 = 4).
In the Arithmetic part of the Constable written test, these skills appear as direct questions ("Find the value of…", "Find the missing number", "Find √5184") and hidden inside other topics: percentage, profit and loss, interest and mensuration questions all end with a simplification step. With no calculator and about 54 seconds per question, fast and accurate simplification saves time across the whole paper.
Core concept
Level 1 — The order of operations (BODMAS)
Work in this order, every time:
- B — Brackets: solve the innermost bracket first, in the order bar (vinculum) → ( ) → { } → [ ].
- O — Of, powers, roots: "of" means multiply; a power or root applies to the number it is attached to. Done before ÷ and ×.
- D, M — Division and Multiplication: equal rank, so work left to right. Do not always do ÷ first.
- A, S — Addition and Subtraction: equal rank, work left to right.
Why "of" is separate from "×": "of" also means multiplication, but it binds the two numbers beside it into one group, so it is done before ÷.
- 36 ÷ 3 of 4 = 36 ÷ 12 = 3
- 36 ÷ 3 × 4 = 12 × 4 = 48
Same numbers, two different answers — watch the word "of".
Why left to right for ÷ and ×: 48 ÷ 6 × 2 = 8 × 2 = 16. Doing 6 × 2 first gives 48 ÷ 12 = 4, which is wrong, because ÷ and × have equal rank.
The vinculum (bar): a line drawn over a group of numbers, such as a bar over "7 − 3", works like a bracket. It is solved before all other brackets.
A minus sign before a bracket changes the sign of every term inside when you open it: 20 − (8 − 3) = 20 − 8 + 3 = 15.
Powers and negative signs: −3² = −(3 × 3) = −9, but (−3)² = (−3) × (−3) = +9. The power attaches only to what is directly under it.
Level 2 — Squares, square roots, cubes and cube roots
Squares from 1 to 30 (memorise):
- 1² to 10²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- 11² to 20²: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
- 21² to 30²: 441, 484, 529, 576, 625, 676, 729, 784, 841, 900
Cubes from 1 to 15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375.
Unit digit of a square depends only on the unit digit of the number:
| Number ends in | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Square ends in | 0 | 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 |
So a perfect square never ends in 2, 3, 7 or 8. Also, a perfect square ending in zeros always has an even number of zeros (100, 3600, 250000).
Square root of a 3- or 4-digit perfect square (unit-digit method):
- Unit digit: if the number ends in 1, the root ends in 1 or 9; 4 → 2 or 8; 9 → 3 or 7; 6 → 4 or 6; 5 → 5; 0 → 0.
- Tens digit: remove the last two digits. Find the largest whole number whose square is less than or equal to what is left. That is the tens digit.
- Choose between the two candidates: square the number ending in 5 that lies between them. If the given number is smaller, take the smaller candidate; if larger, take the larger one.
Prime factor method: pair up equal prime factors and take one from each pair. √1764 = √(2 × 2 × 3 × 3 × 7 × 7) = 2 × 3 × 7 = 42.
Cube root of a perfect cube (up to 6 digits): the unit digit of the cube root is read from the unit digit of the cube: 1→1, 4→4, 5→5, 6→6, 9→9, 0→0, but 2↔8 and 3↔7 swap (a cube ending in 8 has a root ending in 2). For the tens digit, remove the last three digits and find the largest number whose cube is less than or equal to what is left.
Level 3 — Squaring shortcuts, decimals and surds
- Numbers ending in 5: for a number "n5", the square is n × (n + 1) followed by 25. 65² → 6 × 7 = 42 → 4225.
- Near a base (50, 100): use (a ± b)² = a² ± 2ab + b². 98² = (100 − 2)² = 10000 − 400 + 4 = 9604.
- Difference of squares: a² − b² = (a + b)(a − b). 53² − 47² = 100 × 6 = 600.
- Decimals: a decimal (without trailing zeros) can be a perfect square only if it has an even number of decimal places; its root has half as many. √0.0081 = 0.09 (4 places → 2 places); √1.44 = 1.2.
- Fractions: take the root of top and bottom separately. √(169/225) = 13/15.
- Surds (roots that are not whole numbers): take out perfect-square factors. √72 = √(36 × 2) = 6√2. Like surds add like terms: 2√3 + 3√3 = 5√3. Useful values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236.
- Product rule: √a × √b = √(ab). But √(a + b) is not √a + √b: √(9 + 16) = √25 = 5, not 3 + 4 = 7.
- ✓- Order: Brackets → Of / powers / roots → ÷ and × (left to right) → + and − (left to right).
- ✓- Bracket order: bar (vinculum) → ( ) → { } → [ ], innermost first.
- ✓- "of" is done before ÷; "×" is not: 36 ÷ 3 of 4 = 3, but 36 ÷ 3 × 4 = 48.
- ✓- A minus before a bracket flips every sign inside when the bracket is opened.
- ✓- Squares 1–30 and cubes 1–15 must be memorised.
- ✓- A perfect square never ends in 2, 3, 7 or 8, and has an even number of ending zeros.
- ✓- Square root unit digit: 1→1/9, 4→2/8, 9→3/7, 6→4/6, 5→5, 0→0.
- ✓- Cube root unit digit: same as the cube's unit digit, except 2↔8 and 3↔7.
- ✓- n5² = n(n + 1) followed by 25; a² − b² = (a + b)(a − b).
- ✓- √(a + b) ≠ √a + √b; √a × √b = √(ab).
Worked examples
Example 1 (easy). Simplify: 18 + 6 × 4 − 20 ÷ 5
- No brackets, no "of". Do × and ÷ first: 6 × 4 = 24 and 20 ÷ 5 = 4.
- Now 18 + 24 − 4 = 42 − 4 = 38.
Example 2 (easy–medium). Simplify: 48 ÷ 6 × 2 + 60 ÷ 5 of 3
- "of" first: 5 of 3 = 15, so the second part is 60 ÷ 15 = 4.
- First part, left to right: 48 ÷ 6 = 8, then 8 × 2 = 16.
- Total = 16 + 4 = 20.
Example 3 (medium). Simplify: 50 − [20 + {30 − (16 − 3 × 4)} ÷ 2]
- Innermost ( ): 3 × 4 = 12, so (16 − 12) = 4.
- Curly { }: 30 − 4 = 26.
- Inside [ ]: division before addition, 26 ÷ 2 = 13, then 20 + 13 = 33.
- Finally 50 − 33 = 17.
Example 4 (medium). Find √7056.
- Unit digit 6 → the root ends in 4 or 6.
- Remove the last two digits: 70 is left. 8² = 64 ≤ 70 < 81 = 9², so the tens digit is 8. Candidates: 84 or 86.
- 85² = 7225 (8 × 9 = 72, then 25). Since 7056 < 7225, take the smaller: 84.
- Check: 84² = 80² + 2 × 80 × 4 + 4² = 6400 + 640 + 16 = 7056. ✔
Example 5 (medium). Find the value of 53² − 47² + 65².
- 53² − 47² = (53 + 47)(53 − 47) = 100 × 6 = 600.
- 65² = (6 × 7) followed by 25 = 4225.
- Total = 600 + 4225 = 4825.
Example 6 (exam-hard, missing number). ? ÷ 8 × 4 + √625 = 15² − 180
- Right side: 15² = 225, so 225 − 180 = 45.
- √625 = 25. So ? ÷ 8 × 4 + 25 = 45, which gives ? ÷ 8 × 4 = 20.
- Left to right means (? ÷ 8) × 4 = 20, so ? ÷ 8 = 5, and ? = 40.
- Check: 40 ÷ 8 × 4 = 5 × 4 = 20; 20 + 25 = 45. ✔
Example 7 (exam-hard, fractions and roots). Simplify: 2/5 of 3/4 of 1200 + √1296 ÷ 9 × 4
- "of" part: 3/4 of 1200 = 900; 2/5 of 900 = 360.
- √1296 = 36 (36² = 1296). Then left to right: 36 ÷ 9 = 4, and 4 × 4 = 16.
- Total = 360 + 16 = 376.
Example 8 (approximation). 39.97 × 20.02 + √399 ≈ ? (Options: 800, 820, 840, 860)
- Round to friendly numbers: 39.97 ≈ 40, 20.02 ≈ 20, √399 ≈ √400 = 20.
- 40 × 20 + 20 = 820. Answer: 820.
Real-world connection
At a shop, 3 notebooks at ₹40 each and 2 pens at ₹15 each cost 3 × 40 + 2 × 15 = 120 + 30 = ₹150. BODMAS matches common sense: multiply each item first, then add. Blindly working left to right gives (3 × 40 + 2) × 15 = ₹1830, which is absurd.
Squares and roots appear in measurement: a square parade ground of area 2025 m² has a side of √2025 = 45 m, so fencing it needs 4 × 45 = 180 m of wire. Calculators and spreadsheets follow the same order of operations.
- Ignoring order and going left to right for everything: 8 + 2 × 5 is not 50. Multiply first: 8 + 10 = 18.
- Always dividing before multiplying: 48 ÷ 6 × 2 is 16 (left to right), not 48 ÷ 12 = 4.
- Treating "of" like "×": 1/2 ÷ 1/4 of 1/2 = 1/2 ÷ 1/8 = 4. But 1/2 ÷ 1/4 × 1/2 = 2 × 1/2 = 1.
- Forgetting the minus before a bracket: 20 − (8 − 3) = 15, not 20 − 8 − 3 = 9.
- Splitting a root across addition: √(36 + 64) = √100 = 10, not 6 + 8 = 14.
- Wrong decimal places in roots: √0.09 = 0.3 (not 0.03), and √0.0009 = 0.03. A number like 0.9 or 0.081 (odd decimal places) is not the square of a terminating decimal.
- Unit digit elimination: find only the last digit of the answer first. Often only one option matches, and you are done in 10 seconds.
- Perfect-square filter: if an option ends in 2, 3, 7 or 8, it cannot be a perfect square.
- Digit-sum check (casting out nines): add the digits until one digit is left. For 84² = 7056: 8 + 4 = 12 → 3, and 3² = 9; 7 + 0 + 5 + 6 = 18 → 9. They match, so the answer is likely right. This works for +, − and ×.
- Approximation questions: round every number to the nearest friendly value (10, 50, 100, a perfect square) before calculating.
- Missing-number questions: simplify the side without "?" first, then work backwards using inverse operations (÷ becomes ×, + becomes −).
- Mnemonic for bracket order — "Big Roti, Curd, Sambar": Bar → Round ( ) → Curly { } → Square [ ].
How it is asked in the exam
With about 54 seconds per question, aim to finish a simplification question in 30–45 seconds to save time for longer topics.
- Direct value (easy): Find the value of 72 ÷ 8 × 3 + 5.
- "of" and brackets (medium): 84 ÷ 7 of 3 − {6 − (5 − 2)} = ?
- Square, root or cube root (medium): √9216 = ? or ∛13824 = ?
- Missing number (medium–hard): √? + 18 × 2 = 7² + 3
- Approximation (medium): 24.98 × 16.03 − √143 ≈ ?
- Decimals and fractions (medium–hard): √0.0144 + √1.21 = ?
- Comparison (hard): Which is greatest: √3, ∛5 or 1.5?
Easy = one or two operations, no brackets. Medium = "of", nested brackets, or the root of a 4-digit perfect square. Hard = missing number mixed with roots, fractions or decimals.
Wrong options are often built from typical mistakes (doing ÷ before × or ignoring "of"), so a wrong method can land exactly on a printed option. Trust the rule.
- 25 − 15 ÷ 3 × 2 = ? — Answer: 15 (15 ÷ 3 = 5, 5 × 2 = 10, 25 − 10 = 15).
- 72 ÷ 2 of 6 = ? — Answer: 6 (2 of 6 = 12, 72 ÷ 12 = 6).
- √5184 = ? — Answer: 72 (ends in 4 → 2 or 8; 51 → tens digit 7; 75² = 5625 > 5184 → 72).
- 45² = ? — Answer: 2025 (4 × 5 = 20, then 25).
- ∛2744 = ? — Answer: 14 (ends in 4 → 4; remove last three digits → 2, and 1³ ≤ 2 < 8 → 1).
- ✓- Simplify in the order: Brackets → Of / powers / roots → ÷ × (left to right) → + − (left to right).
- ✓- Brackets go bar → ( ) → { } → [ ], innermost first; a minus before a bracket flips the signs inside.
- ✓- "of" is done before ÷; plain "×" is not.
- ✓- Memorise squares 1–30 and cubes 1–15.
- ✓- Perfect squares never end in 2, 3, 7, 8; use the unit digit plus the leftover part to find roots fast.
- ✓- For cube roots, 2↔8 and 3↔7 swap; other unit digits stay the same.
- ✓- Use n5², (a ± b)² and a² − b² shortcuts; √(a + b) ≠ √a + √b.
- ✓- Use unit digits, digit sums and rounding to eliminate options within 54 seconds.
తెలుగు సారాంశం (Telugu summary)
సరళీకరణ (Simplification) అంటే పెద్ద లెక్కను దశలవారీగా ఒకే సంఖ్యగా మార్చడం. దీనికి ఎప్పుడూ BODMAS క్రమాన్ని పాటించాలి.
ముందుగా కుండలీకరణాలు: బార్ (వింక్యులం) → ( ) → { } → [ ] — లోపలి నుంచి బయటికి సాధించాలి.
తర్వాత "of", ఘాతాలు, మూలాలు. "of" అంటే గుణకారమే, కానీ దాన్ని భాగహారం కంటే ముందు చేయాలి.
భాగహారం, గుణకారం రెండింటికీ సమాన ప్రాధాన్యం ఉంది — ఎడమ నుంచి కుడికి చేయాలి. కూడిక, తీసివేత కూడా అలాగే.
బ్రాకెట్ ముందు మైనస్ గుర్తు ఉంటే, బ్రాకెట్ తెరిచినప్పుడు లోపలి గుర్తులన్నీ మారుతాయి.
1 నుంచి 30 వరకు వర్గాలు, 1 నుంచి 15 వరకు ఘనాలు కంఠస్థం చేయండి.
పరిపూర్ణ వర్గం ఎప్పుడూ 2, 3, 7, 8 అంకెలతో ముగియదు.
పెద్ద సంఖ్య వర్గమూలానికి: చివరి అంకెతో ఒకట్ల స్థానం అంకెను, మిగిలిన భాగంతో పదుల స్థానం అంకెను కనుగొనండి.
√(a + b) అనేది √a + √b కి సమానం కాదు అని గుర్తుంచుకోండి.
ప్రతి ప్రశ్నకు సుమారు 54 సెకన్లు మాత్రమే ఉంటాయి; ఒకట్ల స్థానం అంకె, అంకెల మొత్తం సహాయంతో తప్పు ఆప్షన్లను త్వరగా తొలగించండి.
Key terms (English — తెలుగు):
- Simplification — సరళీకరణ
- Bracket — కుండలీకరణం
- Addition / Subtraction — కూడిక / తీసివేత
- Multiplication / Division — గుణకారం / భాగహారం
- Square — వర్గం
- Square root — వర్గమూలం
- Cube — ఘనం
- Cube root — ఘనమూలం
- Perfect square — పరిపూర్ణ వర్గం
- Unit digit — ఒకట్ల స్థానం అంకె
- Approximation — ఉజ్జాయింపు విలువ