Post 15.1 mein 1000 tak ke cubes ka root list se nikaala. Par ∛117649? Itni long list kaun yaad rakhe! Aaj ek kamaal ka rule: last teen digit alag karo, last digit se root ki ikai (sirf ek option!), aur left part se dahai. Bas. Yeh tab chalta hai jab pata ho ki number perfect cube hai (sawaal mein diya ho), aur root 2 digit ka ho (cube 6 digit tak).
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 13824 ek perfect cube hai. ∛13824: kaunsa number teen baar multiply pe 13824?
- 20 × 20 × 20 = 8000: chhota. 30 × 30 × 30 = 27000: bada. To root 20 aur 30 ke beech.
- 25 × 25 × 25: 25 × 25 = 625, 625 × 25 = 15625: bada. 24 × 24 × 24: 24 × 24 = 576, 576 × 24 = 13824: mil gaya! To ∛13824 = 24.
Bade-bade multiply! Ab timer wala sawaal isi tarah try karke kijiye.
Ab phone ya watch mein stopwatch (time measure karne wali watch) chalu kijiye. Neeche wala sawaal isi “try karke” method se kijiye (upar jaisa: estimate lagao, teen baar multiply karke dekho), copy-pencil ke saath. 117649 ek perfect cube hai. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
∛117649 = 49, kyunki 49 × 49 × 49 = 117649
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki compare karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Chahiye: Post 15.1 ki list 1³ se 9³ (1, 8, 27, 64, 125, 216, 343, 512, 729) aur ∛ ka matlab. Comparison ke word (chhota, bada, baith gaya) neeche box mein. Naye word:
📖 Word jo is post mein aayenge
- Last digit (ikai): number ka sabse right wala digit. 117649 ki ikai 9.
- Last teen digit: right se teen digit: 117649 mein 649.
- Left part: last teen digit hata kar jo bacha: 117649 → 117.
- Dahai (root ki): 2-digit root ka pehla digit: 49 mein 4.
- Jagah change karna (swap): 2 ka cube 8 pe khatam, 8 ka cube 2 pe: dono ek doosre ki jagah. Waise hi 3 aur 7.
- Perfect cube hai, pata ho: yeh trick tabhi jab sawaal mein likha ho ki number perfect cube hai.
📏 Bade numbers padhna aur round number mein add karna
Har digit ki ek jagah (place) hoti hai. Right se count karo: pehli jagah Ikai (ones, 1 wale), uske left mein Dahai (tens, 10 wale), phir Saikda (hundreds, 100 wale), phir Hazaar (thousands, 1000 wale).
10 ka bundle: 10 ikai = 1 dahai (10). 10 dahai = 1 saikda (100). 10 saikde = 1 hazaar (1000). Har jagah apne right wali jagah se 10 times badi hai.
| Number | Kaise padhein | Kis cheez se bana |
|---|---|---|
| 98 | attaanve | 9 dahai (90) + 8 ikai |
| 165 | ek sau painsath | 1 saikda (100) + 6 dahai (60) + 5 ikai |
| 999 | nau sau ninyaanve | 9 saikde (900) + 9 dahai (90) + 9 ikai |
| 1011 | ek hazaar gyaarah | 1 hazaar (1000) + 0 saikda + 1 dahai (10) + 1 ikai |
Round number mein chhota number add karna (place box rule): chhote number ko right se likho, yaani ikai ke neeche ikai, dahai ke neeche dahai. Phir har boxes mein jo hai, woh jawab mein utaar do. Ek digit wala number sirf ikai ka zero badalta hai, do digit wala ikai aur dahai dono.
- 100 + 65: 65 mein do digit, to 6 dahai mein aur 5 ikai mein: 1|6|5 = 165.
- 100 + 2: 2 sirf ikai mein, dahai ka 0 waise hi rehta hai: 1|0|2 = 102 (12 nahi, 120 nahi!).
- 200 + 6: 6 sirf ikai mein: 2|0|6 = 206.
- 1000 + 11: 11 mein do digit, to 1 dahai mein aur 1 ikai mein: 1|0|1|1 = 1011.
🔄 Cube root = “reverse cube” (∛ ka matlab)
Cube mein number dete hain: 4 → 4 × 4 × 4 = 64. Cube root mein reverse poochte hain: “Kaunsa number teen baar multiply karne pe 64 aaya?” Jawab 4. Hindi mein ise ghan-mool kehte hain (mool = jad, jahan se ghan uga).
Sign ∛: Chapter 14 wala √ hi, bas uske corners (baayein kandhe) pe chhota 3. ∛64 padho “64 ka cube root”. ∛64 = 4 ka matlab: 4 × 4 × 4 = 64. (Bina 3 ke √ hamesha square root hota hai.)
| Cube (aage) | Cube root (reverse) |
|---|---|
| 2 × 2 × 2 = 8, yaani 2³ = 8 | ∛8 = 2 |
| 3 × 3 × 3 = 27, yaani 3³ = 27 | ∛27 = 3 |
| 5 × 5 × 5 = 125, yaani 5³ = 125 | ∛125 = 5 |
| 10 × 10 × 10 = 1000, yaani 10³ = 1000 | ∛1000 = 10 |
Perfect cube: woh number jo kisi poore number (0, 1, 2, 3 …) ka cube ho: 1, 8, 27, 64, 125 … 100 perfect cube nahi, kyunki 4³ = 64 (chhota) aur 5³ = 125 (bada), beech mein koi whole number nahi.
Check hamesha aasaan: jo root mila, use teen baar likh kar multiply karo. Wahi number wapas aaye to sahi. ∛125 = 5? 5 × 5 = 25, 25 × 5 = 125 ✅.
↔️ Comparison ke word: chhota, bada, “baith gaya”
Is chapter mein comparison hum shabdon mein likhenge, koi sign nahi:
- “64, 117 se chhota hai”: 64 kam hai.
- “125, 117 se bada hai”: 125 zyada hai.
- “64, 117 mein baith gaya” matlab 64, 117 se chhota ya equal hai (117 ke andar aa jaata hai).
- “117, 64 aur 125 ke beech hai”: 117, 64 se bada aur 125 se chhota.
🔁 Chhota sa warm-up
- 4³ aur 5³ kya hain?
- 8³ ki ikai kya hai?
- 117649 ke last teen digit remove karo. Kya bacha?
Answers dekho
- 64 aur 125
- 2 (8³ = 512)
- 117
Teeno aa gaye? Yahi teen cheezein ∛117649 ka jawab hain!
— 📌 charumam.com
🎬 Story: Bhandaare Ke Laddoo
Mohalle ke bhandaare ke liye halwai ne 117649 laddoo banaye aur unhe ek bade cube jaise dher mein jamaana tha: length, chaudai, oonchai, teeno taraf utne hi. Sharma uncle ne poocha: “Har kinaare pe kitne laddoo?”
Zoya ne 5 second mein kaha: “Last digit 9, to root ki ikai 9. Last teen digit remove karo, bacha 117. 4³ = 64 usme baith gaya, 5³ = 125 nahi. To 49!”
Halwai ne 49 × 49 × 49 ka dher banaya: ek bhi laddoo nahi bacha. 🍬 Aaj yahi trick!
— 📌 charumam.com
🪄 Trick Step by Step: Teen Digit Alag, Ikai Ka Add kiya, Left Se Dahai
💡 Is post ki ek badi baat: Jo perfect cube diya hai (jaise 117649, 6 digit tak), uske liye: (1) last teen digit alag karo. (2) Last digit se root ki ikai: 2 ↔ 8, 3 ↔ 7 jagah badalte hain, baaki sab apne jaise. (3) Left part: 1³ se 9³ mein sabse bada cube jo usme baith jaaye, uska number = root ki dahai. Bas, jawab tayyar.
Step 1: Teen digit alag karo
∛117649: last teen digit alag: 117 | 649. (Square root mein do alag karte the, cube root mein teen, kyunki cube mein zero teen-teen increase hote hain: 10³ = 1000.)
Step 2: Last digit se ikai
117649 ki last digit 9. Neeche ki table mein 9 ka cube 729, jo 9 pe hi khatam hota hai. Aur koi digit 9 pe khatam hone wala cube nahi deta. To root ki ikai 9.
| Number ki ikai | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Cube ki ikai | 0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |
| Kaise | 0³ = 0 | 1³ = 1 | 2³ = 8 | 3³ = 27 | 4³ = 64 | 5³ = 125 | 6³ = 216 | 7³ = 343 | 8³ = 512 | 9³ = 729 |
Yaad rakhne ka method: 0, 1, 4, 5, 6, 9 apne jaise. Sirf do add kiye jagah badalte hain: 2 ↔ 8 (2³ = 8, 8³ = 512) aur 3 ↔ 7 (3³ = 27, 7³ = 343). Yaad rakhne ki chhoti madad: 2 + 8 = 10 aur 3 + 7 = 10 (Chapter 3 ke “10 ke dost”). Yeh sirf yaad rakhne ke liye hai; asli reason upar ki table hai (2³ = 8, 8³ = 512).
Step 3: Left parts se dahai
Left 117. List mein dekho: 4³ = 64, 117 mein baith gaya; 5³ = 125, 117 se bada. To sabse bada baithne wala cube 4³, aur root ki dahai 4.
Step 4: Jawab add karo
Dahai 4, ikai 9: ∛117649 = 49.
Step 5: Check (chaho to)
49 × 49: 49 × 40 = 1960 (49 × 4 = 196, peeche 0), 49 × 9 = 441; 1960 + 441 = 2401. Phir 2401 × 49: 2401 × 40 = 96040 (2401 × 4 = 9604, peeche 0), 2401 × 9 = 21609; 96040 + 21609 = 117649. To 49 × 49 × 49 = 117649 ✅
Aur examples, ek ke baad ek
∛19683: last teen digit alag: 19 | 683. Last digit 3 → root ki ikai 7 (3 aur 7 jagah badalte hain). Left 19: 2³ = 8, 19 mein baith gaya; par 3³ = 27, 19 se bada. To root ki dahai 2. Jawab: ∛19683 = 27.
∛438976: last teen digit alag: 438 | 976. Last digit 6 → root ki ikai 6 (6 apne jaisa). Left 438: 7³ = 343, 438 mein baith gaya; par 8³ = 512, 438 se bada. To root ki dahai 7. Jawab: ∛438976 = 76.
∛5832: last teen digit alag: 5 | 832. Last digit 2 → root ki ikai 8 (2 aur 8 jagah badalte hain). Left 5: 1³ = 1, 5 mein baith gaya; par 2³ = 8, 5 se bada. To root ki dahai 1. Jawab: ∛5832 = 18.
Dhyaan: left part 1 ya 2 digit ka bhi ho sakta hai (5832 → 5 | 832). Rule wahi.
🆘 Atak gaye? Aise socho
- Left part: last teen digit remove karo.
- Ikai: 2 ↔ 8, 3 ↔ 7; baaki (0, 1, 4, 5, 6, 9) apne jaise.
- Dahai: 1³ se 9³ mein sabse bada jo left mein baith jaaye.
- Left exactly cube ho (jaise 64 | …): woh bhi “baith gaya” hai, wahi lo.
- Shak ho: ikai check: root ki ikai ka cube, diye hue number ki ikai deta hai? (49: 9³ = 729, ikai 9; 117649 ki ikai bhi 9 ✅)
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Ikai wala rule kyun? 49 × 49 × 49 ki ikai sirf ikaiyon se banti hai (Chapter 14, Post 14.2 jaisa). Pehle 49 × 49: ikai 9 × 9 = 81 ki ikai, yaani 1. Phir us 1 ko 49 ki ikai 9 se multiply: 1 × 9 = 9. To ikai 9, jaise 9³ = 729 ki ikai 9. Matlab cube ki ikai sirf root ki ikai pe tiki hai.
Aur table mein dasoon digit ke cube alag-alag digit pe khatam hote hain. Isliye cube ki ikai dekh kar root ki ikai pakki mil jaati hai, koi do option nahi (square root mein 3 aur 7 dono 9 dete the, cube mein aisa nahi).
Left part dahai kyun deta hai?
40³ = 64000 aur 50³ = 125000 (zero teen times, Post 15.1). 117649 inke beech hai, to root bhi 40 aur 50 ke beech: 40-kuch. “117, 64 aur 125 ke beech” yahi baat hai, bas teen zero hata kar. Isliye last teen digit hatate hain.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: ∛2197 = ?
∛2197: last teen digit alag: 2 | 197. Last digit 7 → root ki ikai 3 (7 aur 3 jagah badalte hain). Left 2: 1³ = 1, 2 mein baith gaya; par 2³ = 8, 2 se bada. To root ki dahai 1. Jawab: ∛2197 = 13.
✔️ Normal method se check: 13 × 13 = 169 (Post 13.2), 169 × 13: 169 × 10 = 1690, 169 × 3 = 507; 1690 + 507 = 2197 ✅
Example 2: ∛54872 = ?
∛54872: last teen digit alag: 54 | 872. Last digit 2 → root ki ikai 8 (2 aur 8 jagah badalte hain). Left 54: 3³ = 27, 54 mein baith gaya; par 4³ = 64, 54 se bada. To root ki dahai 3. Jawab: ∛54872 = 38.
✔️ Normal method se check: 38 × 38: 38 × 30 = 1140 (38 × 3 = 114, peeche 0), 38 × 8 = 304; 1140 + 304 = 1444. Phir 1444 × 38: 1444 × 30 = 43320 (1444 × 3 = 4332, peeche 0), 1444 × 8 = 11552; 43320 + 11552 = 54872. To 38 × 38 × 38 = 54872 ✅
Example 3: ∛262144 = ?
∛262144: last teen digit alag: 262 | 144. Last digit 4 → root ki ikai 4 (4 apne jaisa). Left 262: 6³ = 216, 262 mein baith gaya; par 7³ = 343, 262 se bada. To root ki dahai 6. Jawab: ∛262144 = 64.
✔️ Normal method se check: 64 × 64: 64 × 60 = 3840 (64 × 6 = 384, peeche 0), 64 × 4 = 256; 3840 + 256 = 4096. Phir 4096 × 64: 4096 × 60 = 245760 (4096 × 6 = 24576, peeche 0), 4096 × 4 = 16384; 245760 + 16384 = 262144. To 64 × 64 × 64 = 262144 ✅
Example 4: ∛571787 = ?
∛571787: last teen digit alag: 571 | 787. Last digit 7 → root ki ikai 3 (7 aur 3 jagah badalte hain). Left 571: 8³ = 512, 571 mein baith gaya; par 9³ = 729, 571 se bada. To root ki dahai 8. Jawab: ∛571787 = 83.
✔️ Normal method se check: 83 × 83: 83 × 80 = 6640 (83 × 8 = 664, peeche 0), 83 × 3 = 249; 6640 + 249 = 6889. Phir 6889 × 83: 6889 × 80 = 551120 (6889 × 8 = 55112, peeche 0), 6889 × 3 = 20667; 551120 + 20667 = 571787. To 83 × 83 × 83 = 571787 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Yeh “last digit + pehla part” wali soch CharuMam ka Pehle Dekho (dekh kar jawab) rule hai, jaise Chapter 14 mein square root ke liye. Cube root ke liye yeh aur bhi saaf hai, kyunki ikai ka sirf ek option hota hai. 2 ↔ 8 aur 3 ↔ 7 ke pairs yaad rakhne ke liye Chapter 3 ke “10 se poora” jaise dikhte hain (2 + 8 = 10, 3 + 7 = 10); par yeh sirf yaad rakhne ki madad hai, pairs ka asli reason ikai wali table hai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 6-digit perfect cube (117649) | ✅ Haan | Teen alag, ikai, dahai. |
| 4 ya 5-digit perfect cube (2197, 54872) | ✅ Haan | Left 1 ya 2 digit. |
| 1000 tak ka cube (343) | ✅ List (Post 15.1) | Seedha list. |
| Pata nahi perfect cube hai ya nahi (117650) | ⚠️ Pehle check | Trick koi jawab de degi, par woh galat hoga. Post 15.3. |
| 7-digit ya bada (1030301) | ❌ Is post mein nahi | Root 100 ya usse bada; left part 1000 ya zyada, list chhoti pad jaati hai. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Jagah change karna bhool gaye
Sawaal: ∛54872
❌ Galat: “ikai 2 → root ki ikai 2 → 32.”
✅ Sahi: 2 aur 8 jagah badalte hain: ikai 2 → root ki ikai 8. 54 mein 3³ = 27 baitha → dahai 3 → 38.
Mistake 2: Do digit alag kiye (square root ki habit)
Sawaal: ∛2197
❌ Galat: “21 | 97, left 21…”
✅ Sahi: cube root mein teen: 2 | 197. Left 2 mein 1³ = 1 baitha → dahai 1. Ikai 7 → 3. Jawab 13.
Mistake 3: Left mein bada cube daal diya
Sawaal: ∛117649
❌ Galat: “117 ke paas 125 hai, to dahai 5 → 59.”
✅ Sahi: 125, 117 se bada hai, baithta nahi. Sabse bada baithne wala 64 (4³) → dahai 4 → 49.
Mistake 4: Square root jaisa do option socha
Sawaal: ∛262144
❌ Galat: “ikai 4 → 2 ya 8 → 62 ya 68?”
✅ Sahi: woh square ka rule tha. Cube mein ikai 4 → sirf 4 (4³ = 64). 262 mein 6³ = 216 baitha → 64.
🧠 Rule of thumb: Teen digit alag, last digit se ikai (2 ↔ 8, 3 ↔ 7), left mein sabse bada baithne wala cube se dahai.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 perfect cubes ka cube root. Target: 5 minute (har ek 20 second).
- ∛1728 = ?
- ∛4913 = ?
- ∛10648 = ?
- ∛15625 = ?
- ∛32768 = ?
- ∛42875 = ?
- ∛79507 = ?
- ∛103823 = ?
- ∛148877 = ?
- ∛205379 = ?
- ∛300763 = ?
- ∛389017 = ?
- ∛493039 = ?
- ∛636056 = ?
- ∛941192 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 12
- 17
- 22
- 25
- 32
- 35
- 43
- 47
- 53
- 59
- 67
- 73
- 79
- 86
- 98
Kaise kiya (kuch):
∛79507: last teen digit alag: 79 | 507. Last digit 7 → root ki ikai 3 (7 aur 3 jagah badalte hain). Left 79: 4³ = 64, 79 mein baith gaya; par 5³ = 125, 79 se bada. To root ki dahai 4. Jawab: ∛79507 = 43.
∛636056: last teen digit alag: 636 | 056. Last digit 6 → root ki ikai 6 (6 apne jaisa). Left 636: 8³ = 512, 636 mein baith gaya; par 9³ = 729, 636 se bada. To root ki dahai 8. Jawab: ∛636056 = 86.
— 📌 charumam.com
⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- ∛1331 = ?
- ∛2744 = ?
- ∛3375 = ?
- ∛6859 = ?
- ∛9261 = ?
- ∛24389 = ?
- ∛59319 = ?
- ∛91125 = ?
- ∛175616 = ?
- ∛970299 = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya start! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 11
- 14
- 15
- 19
- 21
- 29
- 39
- 45
- 56
- 99
Dhyaan: 6859: left 6 mein 1³ baitha (2³ = 8 bada), ikai 9 → 19. 970299: left 970 mein 9³ = 729 baitha, ikai 9 → 99.
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har day thoda better, bas yahi chahiye. 💪
— 📌 charumam.com
📝 10 MCQs (Har Option Ka Reason)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. ∛117649 = ?
Jawab aur reason dekho
Sahi jawab: (b) 49
- (a) galat ❌: 94: dahai aur ikai reverse likh di (9 pehle, 4 baad mein).
- (b) sahi ✅: 117 | 649: ikai 9 → 9, 117 mein 4³ = 64 baitha → 49.
- (c) galat ❌: 59: left 117 mein 5³ = 125 baitha samjha, par 125, 117 se bada hai.
- (d) galat ❌: 43: square root wala rule laga diya (ikai 9 → 3 ya 7) aur 3 chuna.
Q2. ∛54872 = ?
Jawab aur reason dekho
Sahi jawab: (a) 38
- (a) sahi ✅: 54 | 872: ikai 2 → 8; 54 mein 3³ = 27 baitha → 38.
- (b) galat ❌: 32: ikai 2 ko 2 hi rakha; 2 aur 8 jagah badalte hain.
- (c) galat ❌: 48: 54 mein 4³ = 64 baitha samjha, par 64, 54 se bada hai.
- (d) galat ❌: 28: 54 mein sabse bada 3³ = 27 baithta hai, par 2³ = 8 le liya.
Q3. Ek perfect cube ki ikai 7 hai. Uske cube root ki ikai kya hogi?
Jawab aur reason dekho
Sahi jawab: (d) 3
- (a) galat ❌: 7: jagah change karna bhool gaye; 7³ = 343 ki ikai 3 hai, 7 nahi.
- (b) galat ❌: Mistake: square root ki tarah do option soche; cube mein har ikai ka sirf ek root-ikai.
- (c) galat ❌: 9: reverse operation kar diya: root dhoondhne ki jagah 7 ka square nikaala (7 × 7 = 49) aur uski ikai 9 likh di. Root ki ikai ka cube 7 pe khatam hona chahiye: 3³ = 27 ✅.
- (d) sahi ✅: 3³ = 27, ikai 7. (3 aur 7 jagah badalte hain.)
Q4. ∛262144 = ?
Jawab aur reason dekho
Sahi jawab: (c) 64
- (a) galat ❌: 46: dahai aur ikai reverse likh di.
- (b) galat ❌: 62: square root wala rule (ikai 4 → 2 ya 8) laga diya aur 2 chuna.
- (c) sahi ✅: 262 | 144: ikai 4 → 4; 262 mein 6³ = 216 baitha → 64.
- (d) galat ❌: 74: 262 mein 7³ = 343 baitha samjha, par 343, 262 se bada hai.
Q5. ∛571787 = ?
Jawab aur reason dekho
Sahi jawab: (b) 83
- (a) galat ❌: 73: 571 mein sabse bada 8³ = 512 baithta hai, par 7³ = 343 le liya.
- (b) sahi ✅: 571 | 787: ikai 7 → 3; 571 mein 8³ = 512 baitha → 83.
- (c) galat ❌: 87: ikai 7 ko 7 hi rakha; 3 aur 7 jagah badalte hain.
- (d) galat ❌: 93: 571 mein 9³ = 729 baitha samjha, par 729, 571 se bada hai.
Q6. Cube root ki is trick mein aakhir se kitne digit alag karte hain?
Jawab aur reason dekho
Sahi jawab: (a) 3
- (a) sahi ✅: Teen, kyunki cube mein zero teen-teen increase hote hain (10³ = 1000).
- (b) galat ❌: 2: square root (Post 14.2) ki habit; wahan 2 alag karte the.
- (c) galat ❌: 1: ikai ke liye sirf last 1 digit dekhte hain, to socha ki alag bhi 1 hi digit karna hai. Par alag teen digit karte hain.
- (d) galat ❌: 4: chaar digit alag kar diye. Tab left part galat ho jaata hai: 117649 → 11 | 7649. Left 11 mein 2³ = 8 baitha (3³ = 27 bada), to dahai 2; ikai 9 → jawab 29. Par 29 × 29 = 841, 841 × 29 = 24389, 117649 nahi. Isliye teen hi alag karo: 117 | 649 → 49.
Q7. ∛2197 = ?
Jawab aur reason dekho
Sahi jawab: (d) 13
- (a) galat ❌: 17: ikai 7 ko 7 hi rakha; 3 aur 7 jagah badalte hain.
- (b) galat ❌: 23: left 2 mein 2³ = 8 baitha samjha, par 8, 2 se bada hai.
- (c) galat ❌: 31: dahai aur ikai reverse likh di.
- (d) sahi ✅: 2 | 197: ikai 7 → 3; 2 mein 1³ = 1 baitha → 13.
Q8. Kaunsa digit kisi perfect cube ki ikai NAHI ho sakta?
Jawab aur reason dekho
Sahi jawab: (c) Aisa koi digit nahi: 0 se 9 sab aa sakte hain
- (a) galat ❌: 3: Mistake: 7³ = 343, ikai 3 aati hai.
- (b) galat ❌: 8: Mistake: 2³ = 8, ikai 8 aati hai.
- (c) sahi ✅: Table: har digit ka cube alag ikai pe khatam, aur dasoon ikai aati hain.
- (d) galat ❌: 2: Mistake: 8³ = 512, ikai 2 aati hai.
Q9. ∛474552 = ?
Jawab aur reason dekho
Sahi jawab: (b) 78
- (a) galat ❌: 68: 474 mein sabse bada 7³ = 343 baithta hai, par 6³ = 216 le liya.
- (b) sahi ✅: 474 | 552: ikai 2 → 8; 474 mein 7³ = 343 baitha → 78.
- (c) galat ❌: 72: ikai 2 ko 2 hi rakha; 2 aur 8 jagah badalte hain.
- (d) galat ❌: 88: 474 mein 8³ = 512 baitha samjha, par 512, 474 se bada hai.
Q10. Cube root ka rule square root (Post 14.2) se aasaan kyun hai?
Jawab aur reason dekho
Sahi jawab: (a) Cube ki ikai se root ki ikai ka sirf ek option milta hai
- (a) sahi ✅: Har ikai alag, to faisle ka step hi nahi chahiye.
- (b) galat ❌: Mistake: cube to square se jaldi bade hote hain (10² = 100, par 10³ = 1000).
- (c) galat ❌: Mistake: 5 wala square (45²) square root mein faisle ke liye tha; cube mein zaroorat nahi.
- (d) galat ❌: Mistake: square root mein ikai ke do option aate the (3 ya 7), cube mein ek.
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🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda mind bhi lagana padta hai. Olympiad ek maths competition (pratiyogita) hota hai.
Q1. ∛681472 = ? Phir check karo (Chapter 13 ki madad se).
Solution dekho
681 | 472. Ikai 2 → 8. Left 681: 8³ = 512 baitha, 9³ = 729 bada → dahai 8. Jawab 88.
Check: 88² Chapter 13 se (base = paas wala round number, yahan 100): 88, base 100 se 12 kam → 88 − 12 = 76 | 12² = 144 → 144 mein 1 saikda left mein: 77 | 44 → 7744. Phir 7744 × 88: 7744 × 80 = 619520 (7744 × 8 = 61952, peeche 0), 7744 × 8 = 61952; 619520 + 61952 = 681472 ✅
Q2. Ek godaam mein 32768 boxes ek bade cube ke shape mein jame hain. Har kinaare pe kitne boxes?
Solution dekho
∛32768: last teen digit alag: 32 | 768. Last digit 8 → root ki ikai 2 (8 aur 2 jagah badalte hain). Left 32: 3³ = 27, 32 mein baith gaya; par 4³ = 64, 32 se bada. To root ki dahai 3. Jawab: ∛32768 = 32.
Check: 32 × 32: 32 × 30 = 960 (32 × 3 = 96, peeche 0), 32 × 2 = 64; 960 + 64 = 1024. Phir 1024 × 32: 1024 × 30 = 30720 (1024 × 3 = 3072, peeche 0), 1024 × 2 = 2048; 30720 + 2048 = 32768. To 32 × 32 × 32 = 32768 ✅
Q3. ∛103823 aur ∛110592 ka difference kitna hai? (Dono perfect cube hain.)
Solution dekho
∛103823: last teen digit alag: 103 | 823. Last digit 3 → root ki ikai 7 (3 aur 7 jagah badalte hain). Left 103: 4³ = 64, 103 mein baith gaya; par 5³ = 125, 103 se bada. To root ki dahai 4. Jawab: ∛103823 = 47.
∛110592: last teen digit alag: 110 | 592. Last digit 2 → root ki ikai 8 (2 aur 8 jagah badalte hain). Left 110: 4³ = 64, 110 mein baith gaya; par 5³ = 125, 110 se bada. To root ki dahai 4. Jawab: ∛110592 = 48.
Difference: 48 − 47 = 1. Yeh do agle cubes hain!
Q4. Ek perfect cube 6 digit ka hai aur 3 pe khatam hota hai. Uske pehle teen digit (left part) 300 se 340 tak ke kisi number jitne hain. Root kya hai?
Solution dekho
Ikai 3 → root ki ikai 7. Left part 300 se 340 tak: 6³ = 216 har baar baith jaata hai, aur 7³ = 343, 340 se bhi bada hai → dahai 6. Root 67.
Check: 67 × 67: 67 × 60 = 4020 (67 × 6 = 402, peeche 0), 67 × 7 = 469; 4020 + 469 = 4489. Phir 4489 × 67: 4489 × 60 = 269340 (4489 × 6 = 26934, peeche 0), 4489 × 7 = 31423; 269340 + 31423 = 300763. To 67 × 67 × 67 = 300763. Left part 300 ✅ (300 se 340 tak mein), ikai 3 ✅.
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Perfect cube ka cube root, ikai ke pattern. |
| Maths Olympiad | 6-digit cube root, cube boxes ke sawaal, ikai ke pattern. |
| NTSE / MAT | Tez cube root, number series. |
| SSC / Bank (aage) | Simplification ke sawaalon mein ∛, 5 second mein. |
Shabdon ka matlab: Olympiad = school ke students ki maths pratiyogita (competition). NTSE = Class 10 ke students ka ek national scholarship (vazifa) exam. MAT = Mental Ability Test, yaani sochne-samajhne ki pariksha (NTSE ka ek part). SSC / Bank exam = badon ke government naukri ke exam. Simplification = long calculation ko chhota karke jawab nikaalna. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “teen alag, ikai ka add kiya, left se dahai.” Ikai table mein sirf do add kiye yaad karne hain: 2 ↔ 8 aur 3 ↔ 7. Baaki sab apne jaise. Aaj 5 perfect cube khud banao (kisi bhi 2-digit number ka cube calculator se) aur dost se root poochho! 🔚
— 📌 charumam.com
⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
117 | 649. Last digit 9 → ikai 9. Left 117: 4³ = 64 baith gaya, 5³ = 125 bada → dahai 4. Jawab 49 ✅ (5 second!)
| Before time | After time | Kitna tez? |
|---|---|---|
| ____ second | ____ second | Before − After = ____ second bache! |
Pehli baar mein time zyada bhi lag sakta hai, ghabraiye nahi! Nayi trick ko 3 se 4 days ki practice chahiye. Phir yeh purane method se kaafi tez ho jaati hai. 🚀
— 📌 charumam.com
🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy kijiye aur CharuMam AI Tutor (tutor.charumam.com) mein paste kar dijiye. Tutor aapko naye sawaal dega aur mistake hone pe samjhayega:
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❓ FAQs (Students Ke Asli Sawaal)
1. Yeh trick har number pe chalti hai?
Sirf perfect cubes pe, aur root 2 digit ka ho (cube 1000 se 999999 tak). Perfect cube na ho to trick galat jawab degi.
2. Square root mein do option the, yahan kyun nahi?
Kyunki dasoon digit ke cube alag-alag digit pe khatam hote hain. Square mein 3² = 9 aur 7² = 49 dono 9 pe khatam the.
3. 2 ↔ 8 kaise yaad rakhoon?
2³ = 8 (8 pe khatam) aur 8³ = 512 (2 pe khatam). Dono ka sum 10. Waise hi 3 + 7 = 10.
4. Left part bilkul cube ho (jaise 64) to?
To wahi cube “baith gaya”, uska number dahai. Jaise 64 | 000 → dahai 4 → 40.
5. Agle post mein kya hai?
Post 15.3: zero wale cubes (∛64000), bade cubes jaise ∛912673, jawab ka check, aur kab yeh trick nahi.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Teen digit alag: 117 | 649.
- Ikai: 2 ↔ 8, 3 ↔ 7, baaki apne jaise.
- Left mein sabse bada baithne wala cube → dahai (117 → 4).
- ∛117649 = 49, sirf ek option, koi decision nahi.
- Trick sirf perfect cube pe (sawaal mein diya ho).
- Shak ho to check: root teen baar multiply.
— charumam.com 🎯
👉 Aage Kya?
Ab Post 15.3: zero wale cubes (∛64000 = 40), 100 ke paas wale bade cubes (∛912673 = 97), jawab ka check, aur woh situation jab yeh trick nahi lagani!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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- 🌐 Website: charumam.com
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