✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 12² = ?
- 36 ke prime factors?
- Kya 50 perfect square?
Answers dekho
- 144
- 2 × 2 × 3 × 3
- Nahi (7² = 49, 8² = 64)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 2 — Square Numbers Ki Properties aur prime factorisation ek baar dekh lijiye.
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🎬 Kahani: Kisan Ka Square Khet
Kisan Ramesh ji ke paas 1156 sq m ka ek square khet hai. Unhe charon taraf baad lagaani hai, to side jaanni hai. 🌾
Unka beta bola: “Area = side × side. To woh number dhoondho jo khud se guna hoke 1156 de!” Par kaise? 30 × 30 = 900, 40 × 40 = 1600… Kya andaaze se hi chalana padega? Nahi! Aaj ek pakka tareeka seekhenge.
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📖 Concept: Square Ka Ulta: Square Root
💡 Is post ki ek badi baat: Square root ulta kaam hai: √y = x jab x² = y. Prime factorisation mein square ke har prime ki jodi hoti hai; har jodi se ek-ek prime uthao, guna karo, root mil gaya. Jodi adhoori? To number perfect square nahi.
Step 1: Square root ka matlab
4² = 16, to 4 ko 16 ka square root kehte hain: √16 = 4.
Do root: (−4)² bhi 16! To 16 ke do integer root: +4 aur −4. Symbol √ hamesha positive wala deta: √16 = 4.
Step 2: Ghatao tareeka (Post 3 se)
81 − 1 − 3 − 5 − 7 − 9 − 11 − 13 − 15 − 17 = 0 → 9 kadam → √81 = 9.
WHY? Pehle n odd ka jod n², to kadam = n. Chhote numbers ke liye badhiya.
Step 3: Prime factorisation: jodi banao
324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 2) × (3 × 3) × (3 × 3).
Har jodi se ek: 2 × 3 × 3 = 18. To √324 = 18.
WHY? Agar y = x × x, to x ke primes y mein do baar aate. Primes ko do barabar dher mein baanto; ek dher = x.
Step 4: Square nahi to?
156 = 2 × 2 × 3 × 13. Jodi: (2 × 2), par 3 aur 13 akele → perfect square nahi.
Square banane ka jugaad: akelon ko jodi do: 156 × 3 × 13 = 156 × 39 = 6084 = 78².
Step 5: Kisan ka khet
1156 = 2 × 2 × 17 × 17 → √1156 = 2 × 17 = 34 m. Baad = 4 × 34 = 136 m.
Check: 34² = 900 + 2 × 30 × 4 + 16 = 1156 ✅
🪜 Atak Gaye? Aise Socho
- Pehle aakhri digit test: 2, 3, 7, 8 pe khatam to root integer nahi.
- Number ko chhote primes se baar-baar bhaag do: 2, 3, 5, 7…
- Primes ko jodi mein ghero.
- Har jodi se ek prime, sab guna.
- Akela bacha? Perfect square nahi.
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🔐 Deep Concept: Kya 2800 Square Hai? Aur Har Prime Ki Power Even
2800 = 2 × 2 × 2 × 2 × 5 × 5 × 7 = 2⁴ × 5² × 7. 7 akela (power 1, odd) → square nahi.
Gehri baat: number perfect square tabhi jab har prime ki power even ho. Kyunki square karne pe har power double hoti: (2² × 3)² = 2⁴ × 3². Yahi niyam Post 1 ke “odd factors” aur Post 2 ke “zero jodi mein” ke peeche bhi hai (10 = 2 × 5, zero bhi jodi mein!).
Sabse chhota guna: 2800 ko square banane ke liye × 7 → 19600 = 140². Bhaag se bhi: 2800 ÷ 7 = 400 = 20².
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✏️ Solved Examples
Example 1 (Easy): √196 prime factorisation se.
196 = 2 × 2 × 7 × 7 → 2 × 7 = 14.
Check: 14 × 14 = 196 ✅
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Example 2 (Medium): Kya 1156 aur 2800 perfect square? (NCERT)
1156 = 2² × 17² → haan, 34. 2800 = 2⁴ × 5² × 7 → nahi.
Check: 52² = 2704, 53² = 2809, 2800 beech mein ✅
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Example 3 (NCERT style): 180 ko kis sabse chhote number se guna karein ki square bane?
180 = 2² × 3² × 5 → × 5 = 900 = 30².
Check: 30 × 30 = 900 ✅
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Ramesh ji ka khet: side 34 m, baad 136 m. Andaaza nahi, pakka hisaab! 🌾
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🏠 Square Root Kahan Kaam Aata
- 🌾 Khet/plot ki side: area se side nikaalna.
- 🖼️ Square photo frame: 400 sq cm ka frame = 20 cm side.
- 🏫 Assembly: 625 bachche square mein: 25 × 25.
- 📐 Pythagoras (aage): tirchhi doori nikaalne mein root.
- 🧩 Puzzle tiles: 144 tukde = 12 × 12 ka square puzzle.
Area pata, side chahiye? Square root lo.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Chhota number (≤ 100) | ✅ Squares ki list ya ghatao | Tez |
| Bada number | ✅ Prime factorisation | Pakka aur saaf |
| Number 2, 3, 7, 8 pe khatam | ❌ Integer root nahi | Square nahi |
| Prime akela bache | ❌ Perfect square nahi | Jodi adhoori |
| Square banana ho | ✅ Akelon se guna/bhaag | Sab jodi poori |
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❌→✅ Worked Mistake
Sawaal: √(2 × 2 × 3 × 3) = ?
❌ Galat: “Sab primes ka guna 36, phir aadha: 18.”
✅ Sahi: har jodi se ek: 2 × 3 = 6. Check: 6² = 36 ✅ (18² = 324)
🧠 Rule of thumb: Root = jodi se ek-ek prime ka guna, aadha karna nahi.
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📝 10 MCQs
Q1. √225 = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 15 × 15 = 225 ✅
- (b) galat: 25 × 25 = 625, galat, isliye yeh galat.
- (c) galat: Aadha karna root nahi hota.
- (d) galat: 5² = 25, 225 nahi, isliye yeh galat.
Q2. 36 ke integer square roots?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Dono ka square 36 ✅
- (b) galat: −6 bhi square karke 36 deta.
- (c) galat: 18 aadha hai, root nahi.
- (d) galat: 18² = 324, galat, isliye yeh galat.
Q3. √144 prime se: 144 = 2⁴ × 3²
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2 × 2 × 3 = 12 ✅
- (b) galat: 24 mein ek 2 zyada liya.
- (c) galat: 6 mein ek 2 kam liya, isliye yeh galat.
- (d) galat: 72 to 144 ka aadha hai.
Q4. Kaunsa perfect square?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Sab powers even ✅
- (b) galat: 2³ ki power odd hai, isliye yeh galat.
- (c) galat: 2 akela hai, odd power.
- (d) galat: 3 akela hai, odd power.
Q5. 72 ko square banane ke liye kis se guna?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 72 = 2³ × 3² → × 2 = 144 ✅
- (b) galat: × 3 se 216, 3 akela ho jaata.
- (c) galat: × 6 se 432, square nahi.
- (d) galat: × 4 se 288, 2 abhi bhi odd.
Q6. √1156 = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2 × 17 = 34 ✅
- (b) galat: 36² = 1296, galat, isliye yeh galat.
- (c) galat: 24² = 576, galat, isliye yeh galat.
- (d) galat: 44² = 1936, galat, isliye yeh galat.
Q7. Ghatao tareeke mein √49 ke kadam?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 49 = 7², 7 kadam ✅
- (b) galat: 49 kadam bahut zyada hain.
- (c) galat: 13 aakhri odd ghataaya, ginti nahi.
- (d) galat: 6 kadam ke baad 13 bachta.
Q8. 2800 square kyun nahi?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 7 ki power 1, odd ✅
- (b) galat: 2⁴, power even hai, isliye yeh galat.
- (c) galat: Zero to 2 hain, even, isliye yeh galat.
- (d) galat: Even hona galat kaaran hai.
Q9. √ symbol deta hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: √16 = 4, positive ✅
- (b) galat: Negative ke liye −√ likhte.
- (c) galat: Symbol sirf ek deta hai.
- (d) galat: Square ulta kaam hai, isliye yeh galat.
Q10. 150 ko square banane ke liye kis se bhaag?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 150 = 2 × 3 × 5² → ÷ 6 = 25 ✅
- (b) galat: ÷ 5 = 30, square nahi.
- (c) galat: ÷ 3 = 50, square nahi.
- (d) galat: ÷ 2 = 75, square nahi.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Negative number ka square root?
Class 8 mein nahi; koi integer khud se guna hoke negative nahi deta.
2. Root aur square ka rishta?
Ulta kaam: √(n²) = n, (√y)² = y.
3. Prime factorisation kaise?
2, 3, 5, 7… se baar-baar bhaag, jab tak 1 na bache.
4. Kya √2 bhi hota?
Haan, par woh integer/fraction nahi; ≈ 1.414. Andaaza: Post 5 — Square Root Ka Andaaza.
5. √(fraction)?
√(9/16) = 3/4. Upar-neeche alag root.
6. Aage kya?
Jo perfect square nahi, unka root ka andaaza: Post 5 — Square Root Ka Andaaza.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy √64 = ?
Solution dekho
8. Check ✅
Q2. Easy √400 = ?
Solution dekho
20. Check: 2 zero → 1 zero ✅
Q3. Easy 49 ke do integer root?
Solution dekho
7, −7. Check ✅
Q4. Easy √(4/25) = ?
Solution dekho
2/5. Check ✅
Q5. Medium √576 prime factorisation se.
Solution dekho
2⁶ × 3² → 2³ × 3 = 24. Check: 24² = 576 ✅
Q6. Medium Kya 288 perfect square?
Solution dekho
2⁵ × 3² → nahi. Check ✅
Q7. Medium √1764 = ?
Solution dekho
2² × 3² × 7² → 42. Check ✅
Q8. Medium Ghatao se √36.
Solution dekho
6 kadam → 6. Check ✅
Q9. NCERT style 252 × ? = square (sabse chhota).
Solution dekho
2² × 3² × 7 → 7 (1764 = 42²). Check ✅
Q10. NCERT style 2925 ÷ ? = square.
Solution dekho
3² × 5² × 13 → 13 (225). Check ✅
Q11. NCERT style Square plot 1024 sq m: side?
Solution dekho
32 m. Check: 2¹⁰ → 2⁵ ✅
Q12. NCERT style 1296 bachche square mein: har qatar?
Solution dekho
36. Check: 2⁴ × 3⁴ → 2² × 3² ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek square ka area 4 guna: side?
Solution dekho
2 guna. Check: √4 = 2 ✅
Q2. n² ke prime factors mein har power even. Kyun?
Solution dekho
Square = powers double. Check: (2 × 3)² = 2² × 3² ✅
Q3. Sabse chhota square jo 6, 9, 15 se divide?
Solution dekho
LCM 90 = 2 × 3² × 5 → × 10 = 900. Check ✅
Q4. √(0.09) = ?
Solution dekho
0.3. Check: 0.3² = 0.09 ✅
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🏆 Olympiad Corner
Q1. √(2⁸ × 5⁴)?
Solution dekho
2⁴ × 5² = 400.
Q2. Sabse chhota n ki 12n square?
Solution dekho
3.
Q3. √(1 + 3 + 5 + … + 49)?
Solution dekho
25.
Q4. 4-digit sabse chhota square?
Solution dekho
1024.
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💗 CharuMam Motivation
Ramesh ji ne andaaze ki jagah jodi banana seekha. Mushkil sawaal ko chhote tukdon (primes) mein todo, sab aasaan! 🌟
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🟢 Master Recap
- 💡 Chaabi: Square root ulta kaam hai: √y = x jab x² = y. Prime factorisation mein square ke har prime ki jodi hoti hai; har jodi se ek-ek prime uthao, guna karo, root mil gaya. Jodi adhoori? To number perfect square nahi.
- √y = x jab x² = y.
- Har square ke do root: + aur −; √ deta +.
- Ghatao: kadam = root.
- Prime jodi se ek-ek = root.
- Akela prime = square nahi.
- Square banane ke liye akelon se guna/bhaag.
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👉 Aage Kya?
250 jaise number ka root? Andaaza: Post 5 — Square Root Ka Andaaza.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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