✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 5 × 5 × 5 = ?
- (−3) × (−3) = ?
- 72 ke prime factors?
Answers dekho
- 125 (5 × 5 = 25, 25 × 5 = 125)
- +9 (minus × minus = plus)
- 2 × 2 × 2 × 3 × 3 (72 → 36 → 18 → 9 → 3 → 1, pehle 2 se, phir 3 se bhaag)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Integers Ka Guna (Sign Rules) aur Post 1 — Exponential Growth Kya Hai? dobara dekh lijiye.
— 📌 charumam.com
🎬 Kahani: Dadaji Ki Lambi Line
Dadaji ne Meenu ki copy mein dekha: “2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2”. 😵
“Beta, itna lamba kyun likha?” Meenu: “Kaagaz 12 baar moda to parat kitni, woh nikaal rahi thi. Par ginti mein ek ‘× 2’ chhoot gaya shayad…”
Dadaji ne pencil li aur likha: 212. “Bas! Yeh bolta hai: 2 ko 12 baar guna karo. Na kuch chhootega, na galat ginoge.”
Meenu: “Wah! Par agar number negative ho, ya 7,200 jaisa bada ho, tab?” Dadaji muskuraaye: “Wahi to aaj seekhna hai.” Post ke end mein Meenu khud 7,200 ko powers mein todegi. ✍️
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📖 Concept: Base, Exponent Aur Unka Khel
💡 Is post ki ek badi baat: Power ek shorthand hai: base batata hai “kaun sa number guna ho raha”, exponent batata hai “kitni baar”. Har sawaal mein pehle yahi do cheezein pehchaano.
Step 1: Likhne aur padhne ka tareeka
$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$
- 3 = base (kaun sa number guna ho raha)
- 5 = exponent / power (kitni baar)
- Padho: “3 raised to the power 5” ya “3 ki power 5”. $3^5$ ko 243 ka exponential form kehte hain.
Khaas naam: $n^2$ = “n squared” (square ka area!), $n^3$ = “n cubed” (cube ka volume!). Yaad hai Book 1 Ch 1 — A Square and a Cube (Notes)?
WHY shorthand? 240 lamba likhoge to 40 baar “× 2”, aur ek bhi chhoota to jawab aadha! Shorthand mein ginti pakki rehti hai.
Step 2: Guna ≠ Power (ek level upar)
| Operation | Matlab | Example |
|---|---|---|
| Guna | baar-baar jodna | $5 \times 3 = 5 + 5 + 5 = 15$ |
| Power | baar-baar guna | $5^3 = 5 \times 5 \times 5 = 125$ |
WHY confusion hota hai? Dono mein “3 baar” hai. Par jaise guna, jod ka shortcut hai, waise hi power, guna ka shortcut hai. Ek seedhi upar.
Step 3: Value nikaalne ke do tareeke
Tareeka A (ek-ek guna): $3^5$: 3 → 9 → 27 → 81 → 243. Har kadam × 3.
Tareeka B (jaane-pehchaane tukde jodo): $2^{10}$ = $2^5 \times 2^5$ = 32 × 32 = 1,024. Ya $3^6 = 3^3 \times 3^3 = 27 \times 27 = 729$.
WHY B sahi? 10 “× 2” ko 5-5 ke do group mein baant diya. Guna ka order/group badalne se jawab nahi badalta.
Kab kaunsa? Power chhoti (≤ 5) → A. Power badi aur even → B (aadha-aadha karo). Agle post Post 3 — Exponent Ke Niyam: Guna Aur Power mein B ka poora niyam.
Step 4: Negative base: sign ka khel
$(-2)^3 = (-2)(-2)(-2) = -8$, par $(-2)^4 = +16$.
- Negative base, even power → positive
- Negative base, odd power → negative
WHY? Minus signs ki jodiyaan banao: har jodi (−)(−) = (+). Even power mein saare minus jodi mein chale jaate hain. Odd mein ek minus akela bach jaata hai.
Isliye $(-1)^{51} = -1$ aur $(-1)^{100} = 1$. Aur $0^n = 0$ (n counting number): 0 ko kitni bhi baar guna, 0 hi.
Step 5: Bade number ko powers mein todo
Tareeka A (baar-baar bhaag): 7,200 ÷ 2 = 3,600 ÷ 2 = 1,800 ÷ 2 = 900 ÷ 2 = 450 ÷ 2 = 225 ÷ 3 = 75 ÷ 3 = 25 ÷ 5 = 5 ÷ 5 = 1.
Tareeka B (factor tree): 7,200 = 72 × 100 = (8 × 9) × (4 × 25) = $2^3 \times 3^2 \times 2^2 \times 5^2$.
$7200 = 2^5 \times 3^2 \times 5^2$
Kab kaunsa? Number mein jaane-pehchaane tukde dikhein (100, 1,000, 72) → B tez. Warna A, sabse chhote prime se shuru, kabhi nahi atakta.
Check: 32 × 9 × 25 = 32 × 225 = 7,200 ✅
Step 6: Letters ke saath bhi
$a \times a \times a \times b \times b = a^3 b^2$ (“a cubed b squared”). $x \times y \times x \times y \times x = x^3 y^2$ (pehle same letters ikattha karo; guna mein order se farak nahi).
WHY letters? Niyam “har number ke liye” likhne ke liye. $n^a$ mein n koi bhi number, a koi bhi ginti.
🪜 Atak Gaye? Aise Socho
- Har power ke liye zor se bolo: “base ko exponent baar guna”. Jaise “3 ko 5 baar guna”.
- Lamba likh ke shuru karo (3 × 3 × 3 × 3 × 3), phir ek-ek guna karo. Shortcut baad mein.
- Negative base? Pehle minus signs gino: even → plus, odd → minus. Phir number nikaalo.
- Prime factors mein atke? Bas 2 se bhaag dete raho jab tak ho sake, phir 3, phir 5, phir 7.
- Jawab check karo: powers ko wapas guna karke asli number milna chahiye.
— 📌 charumam.com
🔐 Deep Concept: Prime Powers = Number Ka DNA (Aur −24 Ka Jaal)
1. DNA wala idea: har number (1 se bada) primes ki powers ka ek hi guna hai. 7,200 ka DNA: $2^5 \times 3^2 \times 5^2$. Isse number ke raaz khul jaate hain:
- Square? Square = kisi number ka “do baar” guna, to har prime do-do ke jode mein aana chahiye → saari powers even. 7,200 mein 25 odd → square nahi. 2 se aur guna karo: 14,400 = $2^6 \times 3^2 \times 5^2$ = 120² ✅
- Cube? Har prime teen-teen ke group mein → saari powers 3 ke multiple. 1,728 = $2^6 \times 3^3$ = 12³ ✅
WHY yeh sach hai? $(2^3 \times 3)^2 = 2^6 \times 3^2$: square karne se har power double hoti hai, to even ho hi jaati hai. Ulta bhi sach.
2. Calculator ka jaal: “−2^4” likho to jawab −16. Kyunki power ka kaam pehle (BODMAS ka “O”), minus baad mein: $-2^4 = -(2^4) = -16$, jabki $(-2)^4 = +16$.
Unifying idea: power hamesha apne base pe lagti hai. Base kya hai, yeh bracket batata hai. Negative number ki power chahiye to bracket zaroor! ⚠️
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Exponential form mein likho: 7 × 7 × 7 × 7 × 7 × 7
Base 7, kitni baar? 6. → 76.
Check: “× 7” gino: 6 baar ✅. Value 73 × 73 = 343 × 343 = 1,17,649.
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Example 2 (Medium): Value nikaalo: (−3)3 × 24
(−3)3: 3 minus, odd → negative: −27. 24 = 16.
−27 × 16 = −432.
Check: 27 × 16 = 27 × 4 × 4 = 108 × 4 = 432, aur ek hi negative factor, to jawab negative ✅
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Example 3 (NCERT style): 2,000 ko prime factors ki powers mein likho. Kya yeh perfect cube hai?
2,000 = 2 × 1,000 = 2 × 23 × 53 = 24 × 53.
Cube ke liye powers 3 ke multiple chahiye; 4 nahi hai → cube nahi.
Check: 16 × 125 = 2,000 ✅. Aur 123 = 1,728, 133 = 2,197: 2,000 beech mein, cube nahi ✅
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Meenu ka jawab: “7,200 = 25 × 32 × 52. Aur yeh square nahi, kyunki 2 ki power odd hai!” Dadaji ne taali bajaayi. 👏
— 📌 charumam.com
🏠 Real Life Mein Powers
- 💾 Computer memory: 1 KB = 210 = 1,024 bytes. Isliye 1 KB “lagbhag” 1,000 hai, exact nahi.
- 📦 Cube ka volume: 4 cm ka dabba = 43 = 64 cm³; 8 cm ka dabba = 83 = 512 cm³ (8 guna!).
- 🌳 Vanshavriksh (family tree): 2 maata-pita, 4 daada-daadi/naana-naani, 8 pardaada… n peedhi pehle 2n purvaj.
- 🧱 Tiles: 5 × 5 ke square floor mein 52 = 25 tiles.
- 🔐 Password: 4 digit PIN ke 104 option (Post 4 — Combinations Aur Powers).
Jahan bhi “same number baar-baar guna” ho, wahan power ka shorthand. Area mein 2, volume mein 3, peedhiyon mein n.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Same number baar-baar guna | ✅ Power mein likho | Chhota aur pakka |
| Alag-alag numbers ka guna (2 × 3 × 5) | ❌ Ek power nahi banti | Base same hona chahiye |
| Negative base ki power | ✅ Bracket lagao: (−2)4 | Warna sign galat |
| Square/cube check karna | ✅ Prime factor powers dekho | Square: sab even; cube: sab 3 ke multiple |
| Baar-baar jodna (5 + 5 + 5) | ❌ Power nahi, guna use karo | 5 × 3 = 15 |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: 43 ki value?
❌ Galat: “43 = 4 × 3 = 12.”
Galti kya hai? Exponent ko seedha guna kar diya. 43 ka matlab 4 ko teen baar guna, 4 ko 3 se guna nahi.
✅ Sahi: 4 × 4 × 4 = 16 × 4 = 64. Check: 43 cube ka volume hai: 4 cm ke dabbe mein 64 chhote 1 cm ke dabbe ✅
Doosri common galti: (−5)2 = −25 likhna. Do minus = plus, to +25. Aur −52 (bina bracket) = −25.
🧠 Rule of thumb: Exponent “kitni baar” hai, “kis se guna” nahi. Base ko exponent baar likho, phir guna karo.
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📝 10 MCQs
Q1. 34 mein base kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 4 exponent hai: woh batata hai kitni baar.
- (b) sahi: 3 hi baar-baar guna ho raha hai, wahi base ✅
- (c) galat: 12 = 3 × 4, iska koi naam nahi aur yeh value bhi nahi.
- (d) galat: 81 poori value hai, base nahi.
Q2. 25 = ?
Solution dekho
Sahi jawab: (c)
- (a) galat: 2 × 5 = 10: exponent ko guna samajh liya.
- (b) galat: 5² = 25: base aur exponent ki jagah ulat di.
- (c) sahi: 2 ko 5 baar guna: 2 × 2 × 2 × 2 × 2 = 32 ✅
- (d) galat: 2 + 5 = 7: jod diya, koi tark nahi.
Q3. (−1)23 = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: +1 tab aata jab power even hoti; 23 odd hai.
- (b) sahi: 23 minus: 11 jodi plus, ek minus bacha → −1 ✅
- (c) galat: Exponent ko hi value maan liya.
- (d) galat: 1 ki koi bhi power 1 hi rehti hai, 23 nahi banti.
Q4. (−2)4 = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: −16 tab aata jab bracket na hota (−2⁴).
- (b) sahi: 4 minus = 2 jodi, sab plus → +16 ✅
- (c) galat: −8 to (−2)³ hai, ek power kam.
- (d) galat: 8 = 2³, power aur sign dono galat.
Q5. a × a × b × b × b = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: a do baar aur b teen baar aaya → a²b³ ✅
- (b) galat: (ab)⁵ mein a bhi 5 baar hota, jo yahan nahi.
- (c) galat: Ginti ulat di: a 2 baar hai, 3 nahi.
- (d) galat: 5ab mein guna ko jod jaisa likh diya.
Q6. 07 = ?
Solution dekho
Sahi jawab: (c)
- (a) galat: Exponent ko value samajh liya.
- (b) galat: 1 tab aata jab power 0 ho (aur base non-zero).
- (c) sahi: 0 ko 7 baar guna karo, jawab 0 hi rahega ✅
- (d) galat: 0 ki positive power bilkul defined hai: 0.
Q7. 360 ka prime power roop?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 8 × 9 × 5 = 360, aur sab base prime ✅
- (b) galat: Value sahi par 10 prime nahi (10 = 2 × 5).
- (c) galat: Value sahi par 45 prime nahi (45 = 3² × 5).
- (d) galat: Value sahi par 6 aur 10 dono prime nahi.
Q8. Kaun sa sabse bada hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2³ = 8 hai, jo 3² = 9 se chhota hai.
- (b) sahi: 3² = 9, baaki sab 8 ya 1 hain, to yahi bada ✅
- (c) galat: 1 ki koi bhi power 1 hi rehti hai.
- (d) galat: 2 × 4 = 8, jo 9 se chhota hai.
Q9. −32 (bina bracket) = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: +9 tab hota jab bracket hota: (−3)².
- (b) sahi: Power sirf 3 pe: −(3²) = −9 ✅
- (c) galat: 3 × 2 = 6: exponent ko guna samajh liya.
- (d) galat: Exponent guna bhi kiya aur sign bhi lagaaya, dono galat.
Q10. Kaun sa number perfect square hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2 ki power 3 odd hai, jodi nahi banti.
- (b) sahi: 4 aur 2 dono even → (2² × 5)² = 20² = 400 ✅
- (c) galat: 3 ki power 3 odd, square nahi (cube hai).
- (d) galat: 2 ki power 1 odd, akela 2 bacha.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Exponent aur power mein farak?
Roz ki bhasha mein dono ek hi. 35 mein “5” exponent hai; poore 35 ko “3 ki 5th power” bhi kehte hain.
2. Kisi number ki power 1 kya hai?
n1 = n. “Ek baar likha” = number khud. Isliye akela 7 bhi 71 hai, yeh niyamon mein kaam aata hai.
3. Power 0 ho to?
Jawab 1 (base 0 na ho to). Kyun? Pattern aur proof Post 5 — Zero Aur Negative Exponent mein.
4. Negative number ki power ka sign kaise yaad rakhein?
Minus ki jodiyaan banao. Even = sab jodi = plus. Odd = ek bacha = minus.
5. Prime factor powers kis kaam ke?
Square/cube check, HCF-LCM, aur bade guna/bhaag aasaan karne ke liye. Yeh number ka DNA hai.
6. Kya 23 aur 32 same hain?
Nahi! 8 aur 9. Base-exponent badalne se value badalti hai. Sirf 24 = 42 = 16 jaisa khaas case barabar hota hai.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Exponential form: 9 × 9 × 9 × 9
Solution dekho
94. Check: “× 9” 4 baar ✅
Q2. Easy Value: 104
Solution dekho
10,000. Check: 1 ke baad 4 zero, power jitne ✅
Q3. Easy Value: (−5)2
Solution dekho
+25. WHY: do minus = ek jodi = plus.
Q4. Easy Exponential form: p × p × q × p
Solution dekho
p teen baar, q ek baar → p3q. Check: kul 4 letters ✅
Q5. Medium 1,000 ko prime powers mein likho.
Solution dekho
1,000 = 103 = (2 × 5)3 = 23 × 53. Check: 8 × 125 = 1,000 ✅
Q6. Medium Value: 23 × 52
Solution dekho
8 × 25 = 200. Check: 200 = 2 × 100 = 2 × 22 × 52 ✅
Q7. Medium (−1)99 + (−1)100 = ?
Solution dekho
−1 + 1 = 0. WHY: 99 odd → −1, 100 even → +1.
Q8. Medium Value: (−4)3
Solution dekho
−64. Check: 43 = 64, aur odd power → minus ✅
Q9. NCERT style Prime powers mein likho: (i) 1,764 (ii) 2,592
Solution dekho
(i) 1,764 = 4 × 441 = 22 × 32 × 72 (ii) 2,592 = 32 × 81 = 25 × 34. Check (i): 4 × 9 × 49 = 1,764 ✅
Q10. NCERT style Value: (−2)3 × (−3)2
Solution dekho
−8 × 9 = −72. Check: ek hi factor negative → jawab negative ✅
Q11. NCERT style Value: 6 × 103 + 4 × 102
Solution dekho
6,000 + 400 = 6,400. Check: 6,400 mein 6 hazaar ke khaane mein, 4 sau ke ✅
Q12. NCERT style Kya 24 × 32 perfect square hai? Square root?
Solution dekho
Haan (powers 4 aur 2 even). √ = 22 × 3 = 12. Check: 122 = 144 = 16 × 9 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek number 2a × 3b hai aur woh 72 hai. a aur b?
Solution dekho
72 = 8 × 9 = 23 × 32 → a = 3, b = 2. WHY sirf yahi? Prime powers ka roop ek hi hota hai (DNA unique).
Q2. (−1)1 + (−1)2 + … + (−1)101 = ?
Solution dekho
Jodiyaan (−1 + 1) = 0 → 50 jodi = 0, aakhri (−1)101 = −1. Jawab −1.
Q3. 25 × 33 ko sabse chhote kis number se guna karein ki perfect square bane?
Solution dekho
Odd powers: 25, 33. Har ek ko even banane ke liye ek-ek 2 aur 3: 6. Result 26 × 34 = (23 × 32)2 = 722 ✅
Q4. Kya kabhi ab = ba ho sakta hai jab a ≠ b (counting numbers)?
Solution dekho
Haan: 24 = 42 = 16. WHY: 4 = 22, to 42 = 24. Counting numbers mein yahi ek jodi hai.
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🏆 Olympiad Corner
Q1. 73 ka units digit?
Solution dekho
7, 49, 343 → 3. Trick: sirf units digit guna karo: 7 → 9 → 3.
Q2. 2n = 4,096. n?
Solution dekho
210 = 1,024, × 4 = 4,096 → n = 12.
Q3. Kaun sa bada: 26 ya 62?
Solution dekho
64 vs 36 → 26. Badi power aksar chhote base pe bhi jeet jaati hai.
Q4. Agar x3 = −125, x?
Solution dekho
Odd power, negative jawab → x negative. (−5)3 = −125 → x = −5.
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Dadaji ki ek chhoti “212” ne Meenu ki poori line bacha di, aur ek chhoota hua “× 2” bhi pakad liya. Maths ke shortcuts aalas ke liye nahi, galti se bachne ke liye bane hain. ✨
Sign wali galtiyan sabse hoti hain, aapse bhi hongi. Har baar bas bracket aur minus ki jodiyaan check karo. Dheere-dheere yeh aadat ban jaayegi! 💪
— 📌 charumam.com
🟢 Master Recap
- 💡 Chaabi: Power ek shorthand hai: base batata hai “kaun sa number guna ho raha”, exponent batata hai “kitni baar”. Har sawaal mein pehle yahi do cheezein pehchaano.
- na = n ko a baar guna; n = base, a = exponent.
- 5 × 3 = 15 (baar-baar jod), 53 = 125 (baar-baar guna).
- Negative base: even power → +, odd power → −.
- −24 = −16, (−2)4 = 16: bracket batata hai base kya hai.
- 0n = 0, n1 = n.
- Prime powers = number ka DNA: 7,200 = 25 × 32 × 52.
- Square: saari prime powers even; cube: saari 3 ke multiple.
— charumam.com 🎯
👉 Aage Kya?
Ab sawaal: 24 × 26 ko bina poora guna kiye kaise likhein? (23)4 kya hai? Agle post mein: exponents ke niyam! ⚙️
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬