Chapter 15 shuru! 🎉 Chapter 14 mein square root seekha: 49 kis number ka square? 7. Aaj ek step aage: cube, yaani number ko teen baar likh kar multiply (4 × 4 × 4 = 64), aur uska reverse cube root (∛64 = 4). Aaj ³ aur ∛ ka matlab, square aur cube ka difference, 1³ se 10³ ki list (agle post ki chaabi), aur round numbers ke cube. Chalo!
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (ek doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). Sawaal: kaunsa number teen baar likh kar multiply karne pe 125 deta hai? Estimates se try karo:
- 3 × 3 × 3: 3 × 3 = 9, 9 × 3 = 27. Bahut chhota.
- 6 × 6 × 6: 6 × 6 = 36, 36 × 6 = 216. Bada.
- 5 × 5 × 5: 5 × 5 = 25, 25 × 5 = 125. Mil gaya! Number 5 hai.
Teen baar try kiya, har baar do times. 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: koi number socho, teen baar multiply karke dekho), copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
7, kyunki 7 × 7 = 49 aur 49 × 7 = 343
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: Chapter 14 ka square (7² = 49) aur √ ka matlab, aur Chapter 6 ka pieces mein multiply (16 × 4 = 40 + 24). Naya: ³ aur ∛ ke sign. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Cube (ghan): number ko teen baar likh kar multiply. 4 ka cube 4 × 4 × 4 = 64.
- ³ (cube ka sign): 4³ padho “4 cube”. Chhota 3 = teen baar likh kar multiply, “× 3” nahi.
- Cube root (ghan-mool): woh number jiska cube diya hua number hai. 64 ka cube root 4.
- ∛ (cube root ka sign): √ jaisa, corners pe chhota 3. ∛64 padho “64 ka cube root”.
- Perfect cube: kisi poore number ka cube: 1, 8, 27, 64, 125 … 100 perfect cube nahi.
- Parat (layer): ek ke upar ek rakhi hui ek tah. Box kai parat se banta hai.
- Kinaara: boxes ki ek seedhi line (edge). Cube boxes ke saare kinaare equal.
📏 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 kya hai? Chhota ³ ka matlab
Chapter 13 aur 14 mein square dekha: number ko usi se ek baar multiply, jaise 4² = 4 × 4 = 16. Ab cube: number ko teen baar likh kar multiply karo. Ise number ke upar-right mein chhota ³ likh kar dikhate hain. 4³ padho “4 ka cube” ya “4 cube”, matlab 4 × 4 × 4.
Teen number ka multiply kaise? Left se, do-do karke: pehle 4 × 4 = 16, phir 16 × 4 = 64. To 4³ = 64. (16 × 4: 10 × 4 = 40 aur 6 × 4 = 24; 40 + 24 = 64.)
Dhyaan: 4³ ka matlab 4 × 3 nahi hai (woh sirf 12 hota)! Aur 4² (16) bhi nahi. Chhota ³ batata hai ki 4 ko kitni baar likh kar multiply karna hai: teen baar.
Name “cube” kyun? Cube ka matlab ghan, yaani box jiski length, chaudai aur oonchai teeno equal hon: jaise ludo ka paasa (dice) ya chini ka dhela. 4 × 4 = 16 chhote blocks ki ek parat (layer) banao, aur aisi 4 parat ek ke upar ek rakho: 16 + 16 + 16 + 16 = 64 blocks ka box. Isliye 4³ = 64.
| Square (²) | Cube (³) | |
|---|---|---|
| Kitni baar likh kar multiply | 2 baar: 4 × 4 | 3 baar: 4 × 4 × 4 |
| 4 ke liye jawab | 16 | 64 |
| Shakal | square (flat, jaise rangoli) | box (jaise paasa) |
| Reverse | square root √ (Chapter 14) | cube root ∛ (aaj) |
🔄 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 ✅.
🔁 Chhota sa warm-up
- 4 × 4 = ?
- 16 × 4 = ?
- 7² = ?
Answers dekho
- 16
- 64 (10 × 4 = 40, 6 × 4 = 24, 40 + 24 = 64)
- 49 (7 × 7)
Pehle do mila do: 4 × 4 × 4 = 64. Yahi 4³ hai!
— 📌 charumam.com
🎬 Story: Chini Ke Dhelon Ka Box
Dadi ke place Kabir ko ek chhota box mila jisme 64 chini ke dhele (chhote cube) bilkul fit the: utni hi length, utni hi chaudai, utni hi oonchai. Dadi ne poocha: “Bata, har kinaare pe kitne dhele?”
Kabir ne pehle socha 8 (kyunki 8 × 8 = 64). Par box to teen taraf se equal tha! Usne neeche ki parat count ki: 4 line, har line mein 4, yaani 16. Aur aisi 4 parat: 16 + 16 + 16 + 16 = 64.
“Har kinaare pe 4, Dadi! 4 × 4 × 4 = 64.” Dadi muskuraayi: “Isi ko cube kehte hain, aur reverse poochna cube root.” 🧊
— 📌 charumam.com
🪄 Trick Step by Step: List Banao, Reverse Padho, Zero Teen-Teen
💡 Is post ki ek badi baat: Cube = number teen baar likh kar multiply (left se do-do karke: 4 × 4 = 16, 16 × 4 = 64). Cube root = reverse: ∛64 = 4. 1³ se 10³ ki list ek baar bana lo; phir cube root bas list ko reverse padhna hai. Round number mein zero teen times (20³ = 8000), aur reverse mein teen zero ka ek (∛8000 = 20).
Step 1: Cube = square × number
Square aapko aata hai (Chapter 14). Cube = square, phir ek baar aur wahi number se multiply. 6³: 6 × 6 = 36 (square), phir 36 × 6: 30 × 6 = 180 aur 6 × 6 = 36; 180 + 36 = 216.
Step 2: 1³ se 10³ ki list
Isi tarah 1 se 10 tak sab cube banao. Yeh list is chapter ki chaabi hai, ise ek baar khud copy pe banao:
| Cube | Matlab | Kaise (left se, do-do karke) | Perfect cube |
|---|---|---|---|
| 1³ | 1 × 1 × 1 | 1 × 1 = 1, 1 × 1 = 1 | 1 |
| 2³ | 2 × 2 × 2 | 2 × 2 = 4, 4 × 2 = 8 | 8 |
| 3³ | 3 × 3 × 3 | 3 × 3 = 9, 9 × 3 = 27 | 27 |
| 4³ | 4 × 4 × 4 | 4 × 4 = 16, phir 16 × 4: 10 × 4 = 40 aur 6 × 4 = 24; 40 + 24 = 64 | 64 |
| 5³ | 5 × 5 × 5 | 5 × 5 = 25, phir 25 × 5: 20 × 5 = 100 aur 5 × 5 = 25; 100 + 25 = 125 | 125 |
| 6³ | 6 × 6 × 6 | 6 × 6 = 36, phir 36 × 6: 30 × 6 = 180 aur 6 × 6 = 36; 180 + 36 = 216 | 216 |
| 7³ | 7 × 7 × 7 | 7 × 7 = 49, phir 49 × 7: 40 × 7 = 280 aur 9 × 7 = 63; 280 + 63 = 343 | 343 |
| 8³ | 8 × 8 × 8 | 8 × 8 = 64, phir 64 × 8: 60 × 8 = 480 aur 4 × 8 = 32; 480 + 32 = 512 | 512 |
| 9³ | 9 × 9 × 9 | 9 × 9 = 81, phir 81 × 9: 80 × 9 = 720 aur 1 × 9 = 9; 720 + 9 = 729 | 729 |
| 10³ | 10 × 10 × 10 | 10 × 10 = 100, 100 × 10 = 1000 (peeche ek 0 aur) | 1000 |
Step 3: Cube root: list reverse padho
∛343 = ? List ke last boxes mein 343 dhoondho: 7³ = 343. To ∛343 = 7. ∛729 = ? 9³ = 729, to 9.
Step 4: Round number: zero teen times
40³: 4³ = 64, aur 40 ke ek 0 ke badle teen 0: 64000. Reverse: ∛64000: teen 0 remove karo, bacha 64, ∛64 = 4, ek 0 waapas: 40. (Kyun, neeche box mein.)
0️⃣ Round number ka cube aur cube root: zero teen times, zero ek-tihai
Cube mein zero teen times: 20³ = 20 × 20 × 20. Pehle digit: 2 × 2 × 2 = 8. Har 20 ke saath ek 0 tha, aur teen 20 hain, to teen 0: 8000. Isi tarah 30³: 3 × 3 × 3 = 27, peeche teen 0, yaani 27000.
Kyun? 20 = 2 × 10. To 20 × 20 × 20 mein teen 2 aur teen 10 hain. Multiply mein jagah badal sakte hain (Chapter 7 ka rule), to pehle 2 × 2 × 2 = 8, phir 10 × 10 × 10 = 1000. Aur 8 × 1000 = 8000 (1000 se multiply = peeche teen 0).
Cube root mein reverse, zero ek-tihai (teen parts mein ek): ∛8000: peeche ke teen 0 remove karo, bacha 8. ∛8 = 2. Teen 0 ka ek-tihai, ek 0: 20. Check: 20 × 20 × 20 = 8000 ✅.
Dhyaan: zero teen-teen ke jhund mein (3, 6, 9 …) hone chahiye. 27000 (teen 0) ✅ → 30. Par 2700 (do 0) ya 270 (ek 0) kisi round number ka cube nahi, kyunki round number ka cube hamesha teen-teen zero badhaata hai.
Step 5: Check karo
Root mila? Use teen baar multiply karo. ∛216 = 6? 6 × 6 = 36, 36 × 6 = 216 ✅.
List mein ek mazedaar baat (agle post ka secret)
List ke cubes ki last digit dekho: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0. Saari alag! Koi do cube ek hi digit pe khatam nahi hote. Square mein aisa nahi tha (Chapter 14: 3² = 9 aur 7² = 49, dono 9 pe). Post 15.2 mein yahi secret bade cubes ka root 5 second mein dega.
🆘 Atak gaye? Aise socho
- ³ ka matlab: teen baar likh kar multiply, “× 3” nahi.
- Teen number ka multiply: pehle do ka (4 × 4 = 16), phir teesre se (16 × 4 = 64).
- ∛: list mein cube dhoondho, opposite wala number root.
- Zero: cube mein har 0 ke teen 0; root mein teen 0 ka ek.
- Shak ho to: root ko teen baar multiply karke check.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Square “square” hai, cube “box”. 3² = 9: ek flat parat, 3 line aur har line mein 3 tiles. 3³ = 27: aisi hi 3 parat ek ke upar ek, har parat mein 9 chhote cube: 9 + 9 + 9 = 27. Teesra 3 batata hai “kitni parat”.
Isliye cube = square × number: 3³ = 3² × 3 = 9 × 3 = 27. Neeche chitra mein dono saath-saath.
Cube root kyun “reverse”?
Cube mein kinaara diya, count karo kitne chhote cube: kinaara 4 → 64. Cube root mein total cube diye, poochho kinaara kitna: 64 → 4. Wahi box, bas sawaal reverse. Isliye ∛ ka jawab hamesha check ho sakta hai: kinaara teen baar multiply karo.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 6³ = ?
Pehle square: 6 × 6 = 36. Phir 36 × 6: 30 × 6 = 180 aur 6 × 6 = 36; 180 + 36 = 216.
✔️ Normal method se check: 216 ko 6 parat mein socho, har parat 36: 36 + 36 = 72, 72 + 36 = 108, 108 + 36 = 144, 144 + 36 = 180, 180 + 36 = 216 ✅
Example 2: ∛512 = ?
List mein 512 dhoondho: 8³ = 512. To ∛512 = 8.
✔️ Normal method se check: 8 × 8 = 64, 64 × 8: 60 × 8 = 480 aur 4 × 8 = 32; 480 + 32 = 512 ✅
Example 3: 30³ = ?
3³ = 27. 30 mein ek 0, cube mein teen 0: 27000.
✔️ Normal method se check: 30 × 30 = 900 (Post 13.3), 900 × 30 = 27000 (9 × 3 = 27, peeche teen 0) ✅
Example 4: Ek cube box 1000 chhote cubes se bana. Har kinaare pe kitne cube?
Kinaara = ∛1000. List: 10³ = 1000. To har kinaare pe 10 cube.
✔️ Normal method se check: 10 × 10 = 100 (ek parat), 100 × 10 = 1000 (10 parat) ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Hindi mein cube ko ghan aur cube root ko ghan-mool kehte hain. Yahan bhi pehla step yahi hai: 1 se 9 tak ke cubes mann mein pakke karna. Ek aur baat: square ki tarah, base ke paas wale numbers (jaise 98, 103) ka cube nikaalne ki bhi ek tez method hoti hai; Chapter 13 ka “base se distance” wala idea wahan bhi kaam aata hai. Hum is chapter mein reverse, yaani ghan-mool, tez nikaalna seekhenge.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 1 se 1000 tak ka perfect cube (343) | ✅ Haan, list | Seedha list reverse padho. |
| Round number ka cube (40³) | ✅ Zero teen times | 4³ = 64, peeche teen 0. |
| Peeche 000 wala perfect cube (64000) | ✅ Teen 0 ka ek | ∛64 = 4 → 40. |
| Perfect cube nahi (100) | ❌ Poora root nahi | 4³ = 64 aur 5³ = 125 ke beech. |
| Bada perfect cube (117649) | ⏭️ Post 15.2 | Last digit wala rule. |
| Sirf “square” poocha (4²) | ❌ Cube nahi | 4² = 16, do baar multiply. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: ³ ko × 3 samjha
Sawaal: 4³
❌ Galat: “4 × 3, to 12.”
✅ Sahi: 4³ = 4 × 4 × 4 = 64. Chhota 3 matlab teen baar likh kar multiply.
Mistake 2: Square kar diya
Sawaal: 5³
❌ Galat: “5 × 5, to 25.” (Yeh 5² hai.)
✅ Sahi: ek baar aur 5 se multiply: 25 × 5 = 125.
Mistake 3: Cube root mein 3 se parts kiye
Sawaal: ∛27
❌ Galat: “27 ke 3 equal parts, har part 9.”
✅ Sahi: kaunsa number teen baar multiply pe 27? 3 × 3 × 3 = 27, to 3. Check: 9 × 9 × 9 to 729 hota!
Mistake 4: Zero galat count kiye
Sawaal: 20³
❌ Galat: “2³ = 8, ek 0 lagao, 80” ya “do 0, 800”.
✅ Sahi: teen 20 hain, har ek ka ek 0: teen 0 → 8000.
🧠 Rule of thumb: Cube = teen baar multiply (left se do-do), cube root = list reverse, zero cube mein teen times aur root mein teen ka ek, aur hamesha check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: cube ya cube root, list se ya mann se. Target: 4 minute.
- 2³ = ?
- 5³ = ?
- 9³ = ?
- ∛27 = ?
- ∛216 = ?
- 10³ = ?
- ∛729 = ?
- 4³ = ?
- ∛1 = ?
- 7³ = ?
- ∛512 = ?
- 20³ = ?
- ∛64000 = ?
- ∛343 = ?
- 3³ = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 8
- 125
- 729
- 3
- 6
- 1000
- 9
- 64
- 1
- 343
- 8
- 8000
- 40
- 7
- 27
Kaise kiya (kuch): 9³: 9 × 9 = 81, 81 × 9: 80 × 9 = 720 aur 1 × 9 = 9; 720 + 9 = 729. ∛64000: teen 0 remove karo, ∛64 = 4, ek 0 waapas → 40. 20³: 2³ = 8, teen 0 → 8000.
— 📌 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.
- 1³ = ?
- 6³ = ?
- 8³ = ?
- 30³ = ?
- ∛8 = ?
- ∛125 = ?
- ∛1000 = ?
- ∛27000 = ?
- ∛8000 = ?
- ∛64 = ?
| 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
- 1
- 216
- 512
- 27000
- 2
- 5
- 10
- 30
- 20
- 4
Dhyaan: 8³ = 512 (8 × 8 = 64, 64 × 8 = 512). 30³ = 27000 (27, teen 0). ∛27000 = 30, ∛8000 = 20 (teen 0 ka ek 0).
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. 4³ = ?
Jawab aur reason dekho
Sahi jawab: (a) 64
- (a) sahi ✅: 4 × 4 = 16, 16 × 4 = 64.
- (b) galat ❌: 12: ³ ko “× 3” samjha: 4 × 3 = 12.
- (c) galat ❌: 16: square kar diya (4 × 4), teesra 4 bhool gaye.
- (d) galat ❌: 43: 4 aur 3 ko saath-saath likh diya.
Q2. ∛27 = ?
Jawab aur reason dekho
Sahi jawab: (d) 3
- (a) galat ❌: 9: 27 ke 3 equal parts kar diye (9 + 9 + 9 = 27); cube root parts karna nahi hai.
- (b) galat ❌: 81: 27 ko 3 se multiply kar diya (27 × 3 = 81).
- (c) galat ❌: 5: list ki galat line padh li; 5³ = 125 hai, 27 nahi.
- (d) sahi ✅: 3 × 3 = 9, 9 × 3 = 27.
Q3. 10³ = ?
Jawab aur reason dekho
Sahi jawab: (c) 1000
- (a) galat ❌: 100: square kar diya (10 × 10), teesra 10 bhool gaye.
- (b) galat ❌: 10000: teen ki jagah chaar 0 laga diye.
- (c) sahi ✅: 10 × 10 = 100, 100 × 10 = 1000.
- (d) galat ❌: 30: ³ ko “× 3” samjha: 10 × 3 = 30.
Q4. ∛1000 = ?
Jawab aur reason dekho
Sahi jawab: (b) 10
- (a) galat ❌: 1: saare 0 hata diye, ek bhi waapas nahi lagaya.
- (b) sahi ✅: Teen 0 ka ek 0, aur ∛1 = 1 → 10.
- (c) galat ❌: 100: teen 0 mein se do 0 rakh liye; ek-tihai (teen parts mein ek) ki jagah do-tihai (teen parts mein do) le liya.
- (d) galat ❌: 333: 1000 ke teen equal parts karne ki koshish (333 + 333 + 333 = 999).
Q5. Kaunsa perfect cube hai?
Jawab aur reason dekho
Sahi jawab: (a) 125
- (a) sahi ✅: 5 × 5 × 5 = 125.
- (b) galat ❌: Mistake: 100 round dikhta hai, par 4³ = 64 aur 5³ = 125 ke beech hai.
- (c) galat ❌: Mistake: 150, 5³ = 125 aur 6³ = 216 ke beech hai.
- (d) galat ❌: Mistake: 200 bhi 125 aur 216 ke beech hai.
Q6. 7³ = ?
Jawab aur reason dekho
Sahi jawab: (d) 343
- (a) galat ❌: 49: square kar diya (7 × 7), teesra 7 bhool gaye.
- (b) galat ❌: 21: ³ ko “× 3” samjha: 7 × 3 = 21.
- (c) galat ❌: 243: 49 × 7 mein 40 × 7 = 280 ki jagah 180 likh diya: 180 + 63 = 243.
- (d) sahi ✅: 7 × 7 = 49; 49 × 7: 40 × 7 = 280, 9 × 7 = 63; 280 + 63 = 343.
Q7. 20³ = ?
Jawab aur reason dekho
Sahi jawab: (c) 8000
- (a) galat ❌: 60: ³ ko “× 3” samjha: 20 × 3 = 60.
- (b) galat ❌: 400: square kar diya (20 × 20).
- (c) sahi ✅: 2³ = 8, teen 0 → 8000.
- (d) galat ❌: 800: sirf do 0 lagaye.
Q8. ∛512 = ?
Jawab aur reason dekho
Sahi jawab: (b) 8
- (a) galat ❌: 9: list mein ek line neeche padh li; 9³ = 729.
- (b) sahi ✅: 8 × 8 = 64, 64 × 8 = 512.
- (c) galat ❌: 256: 512 ka half kar diya (256 + 256 = 512).
- (d) galat ❌: 7: list mein ek line upar padh li; 7³ = 343.
Q9. Ek cube box 64 chhote cubes se bana. Har kinaare pe kitne cube?
Jawab aur reason dekho
Sahi jawab: (a) 4
- (a) sahi ✅: 4 × 4 × 4 = 64.
- (b) galat ❌: 8: square root le liya (8 × 8 = 64), par box teen taraf se equal hai.
- (c) galat ❌: 16: sirf ek parat ke cube count kar liye (4 × 4 = 16).
- (d) galat ❌: 32: 64 ka half kar diya.
Q10. 2³ aur 3² mein kaun bada?
Jawab aur reason dekho
Sahi jawab: (d) 3² (9), 2³ (8) se bada
- (a) galat ❌: Mistake: upar ka chhota number sirf batata hai kitni baar multiply; dono ko nikaal kar compare karo.
- (b) galat ❌: Mistake: 2 × 3 = 6 aur 3 × 2 = 6 samjha, jo ³ aur ² ka matlab nahi.
- (c) galat ❌: Mistake: 8 aur 9 ki comparison reverse kar di.
- (d) sahi ✅: 2³ = 2 × 2 × 2 = 8 aur 3² = 3 × 3 = 9. 9 bada.
— 📌 charumam.com
🧠 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. 1³ + 2³ + 3³ kitna hai? Ise (1 + 2 + 3)² se milao. Kya dikha?
Solution dekho
1³ + 2³ + 3³ = 1 + 8 + 27 = 36. Aur 1 + 2 + 3 = 6, aur 6² = 36. Equal!
Aur aage: 1³ + 2³ + 3³ + 4³ = 36 + 64 = 100, aur 1 + 2 + 3 + 4 = 10, 10² = 100. Pehle kuch cubes ka sum = pehle kuch numbers ke sum ka square. Mazedaar pattern! 🤩
Q2. 27 chhote cubes se 3 × 3 × 3 ka bada box banaya aur bahar se poora colour diya. Kitne chhote cubes pe bilkul colour nahi laga?
Solution dekho
Colour sirf bahar ki parat pe laga. Bahar ki parat har taraf se remove karo: ek taraf se 1 aur opposite wali taraf se 1 subtract karo, to kinaara 3 se 1 reh jaata hai (3 − 1 − 1 = 1). Andar bacha 1 × 1 × 1 = 1 cube, bilkul beech wala.
Q3. 5³ − 4³ = ? (Do agle cubes ka difference)
Solution dekho
5³ = 125, 4³ = 64. 125 − 64 = 61.
Check: 64 + 61 = 125 ✅. (Socho: 4 × 4 × 4 ke boxes ko 5 × 5 × 5 banane ke liye 61 naye cube chahiye.)
Q4. Kaunsa whole number apne hi cube ke equal hai?
Solution dekho
1 × 1 × 1 = 1, aur 0 × 0 × 0 = 0. To 0 aur 1. 2 se aage cube bahut tez increase hota hai: 2³ = 8, 3³ = 27.
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🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Cube aur cube root ka matlab, 1 se 10 ke cubes, perfect cube pehchanna. |
| Maths Olympiad | Cubes ka pattern (1³ + 2³ + 3³ = 6²), colour wale boxes ke sawaal. |
| NTSE / MAT | Number series mein cubes: 1, 8, 27, 64, … agla 125. |
| SSC / Bank (aage) | ∛ wale long sawaal ko chhota karna, 5 second mein list se. |
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. Number series = numbers ki ek line jo kisi rule se aage increase hoti hai, aur agla number batana hota hai. Jaise 5, 10, 15, 20, … mein har baar 5 jud raha hai, to agla 25. Class 6–8 = school ki chhathi se aathvi kaksha.
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👩🏫 CharuMam Ka Tip
Mantra: “teen baar multiply, list reverse, zero teen-teen.” Aaj 1³ se 10³ ki list apni copy ke pehle panne pe khud likho aur daily ek baar zor se padho: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Agle post mein yahi list 117649 jaise bade cube ka root 5 second mein degi! 🧊
— 📌 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)
1³ se 10³ ki list mein dekho: 7³ = 343. To jawab 7, yaani ∛343 = 7 ✅ (Ek bhi try nahi!)
| 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:
— 📌 charumam.com
❓ FAQs (Students Ke Asli Sawaal)
1. Cube aur square mein kya difference hai?
Square = 2 baar likh kar multiply (4² = 16, square). Cube = 3 baar likh kar multiply (4³ = 64, box).
2. 4³ ko 4 × 3 kyun nahi likh sakte?
Chhota 3 “kitni baar likh kar multiply” batata hai, “kisse multiply” nahi. 4 × 3 = 12 hota hai, 4³ = 64.
3. List yaad karna zaroori hai?
Haan, 1³ se 10³ (sirf 10 number). Agle post ki trick isi pe tiki hai.
4. Har number ka cube root poora hota hai?
Nahi. 100 ka cube root 4 aur 5 ke beech hai (4³ = 64, 5³ = 125). Sirf perfect cubes ka root whole number.
5. Agle post mein kya hai?
Post 15.2: 117649 jaise 6-digit perfect cube ka root, last digit se: ∛117649 = 49.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Cube = teen baar likh kar multiply: 4³ = 4 × 4 × 4 = 64.
- Square square (2 baar), cube box (3 baar).
- Cube root = reverse cube: ∛64 = 4.
- 1³ se 10³: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- Round number: zero teen times (20³ = 8000); root mein teen ka ek (∛8000 = 20).
- Cubes ki last digit sab alag (agle post ka secret).
— charumam.com 🎯
👉 Aage Kya?
Ab bade cubes: Post 15.2 mein ∛117649 = 49 jaise sawaal, sirf last digit aur left part dekh kar, bina long multiply ke!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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- 🌐 Website: charumam.com
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