Power Play Ke Tricky Sawaal | Class 8 Maths Book 1 Ch 2

📘 Class 8 · Ganita Prakash Book 1Ch 2 · Power PlayPost 9 / 9⏱ 25–30 min

Power Play Ke Tricky Sawaal 🧩

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

2⁵ × 2³ = 2^?
(3²)⁴ = 3^?
5⁰ = ?

Answers dekho

8 (exponents jodo)
8 (exponents guna)
1

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 3 — Exponent Ke Niyam: Guna Aur Power aur Post 5 — Zero Aur Negative Exponent dobara dekh lijiye.

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🎬 Kahani: Meenu Ki Chunauti: 2³⁰ Ya 3²⁰?

Meenu ne board pe likha: 2³⁰ aur 3²⁰ — kaun bada? Shart: calculator nahi, aur poora number nikaalna bhi nahi! 😈

Golu: “2³⁰! Exponent 30 bada hai na.” Meenu: “Aur 3 base bada hai…”

Dono ke paas ek-ek “taakat” hai. Is post mein hum aise hi tricky sawaalon ke pakke tareeke seekhenge, aur yeh chunauti bhi jeetenge. 🏆

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📖 Concept: Tricky Sawaal Ke 4 Hathiyaar

💡 Is post ki ek badi baat: Powers ke tricky sawaal mein pehle same base ya same exponent banao. Jab ek cheez barabar ho, to doosri dekh ke faisla turant. Exponent ke niyam isi ke auzaar hain.

Step 1: Hathiyaar 1: Compare karna

Tareeka A (same exponent): 2³⁰ = (2³)¹⁰ = 8¹⁰, aur 3²⁰ = (3²)¹⁰ = 9¹⁰. Ab exponent same (10), base 9 > 8 → 3²⁰ bada ✅

Tareeka B (same base): 8⁵ vs 4⁷: 8⁵ = 2¹⁵, 4⁷ = 2¹⁴ → 8⁵ bada.

WHY kaam karta hai? Same base (1 se bada) → bada exponent = zyada baar guna = bada number. Same exponent → bada base har baar bada guna = bada number.

Kab kaunsa? Base ek doosre ki power hon (2, 4, 8, 16) → Tareeka B. Exponents ka common factor ho (30, 20 → 10) → Tareeka A.

Step 2: Hathiyaar 2: Units digit ka chakra

7 ki powers ke aakhri ank: 7, 49, 343, 2401, 16807… → 7, 9, 3, 1, 7, 9, 3, 1…

WHY chakra? Units digit sirf pichle units digit pe depend karta hai (7 × 1 = 7 phir se shuru). Sirf 10 possible ank, to chakra banna hi hai.

7⁴⁵: chakra lambaai 4; 45 = 4 × 11 + 1 → 1st position → 7.

Base ka units	Chakra	Lambaai
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
4	4, 6	2
9	9, 1	2
0, 1, 5, 6	wahi rehta	1

Step 3: Hathiyaar 3: Hamesha / Kabhi-kabhi / Kabhi nahi

“a^m × aⁿ = a^m+n” → hamesha (gin ke saabit).
“2ⁿ even hai” → kabhi-kabhi: n ≥ 1 pe haan, par 2⁰ = 1 odd!
“(a + b)² = a² + b²” → kabhi-kabhi (sirf jab a ya b = 0). Kyun? Book 1 Ch 6 — (a + b)² Ka Formula mein.
“5ⁿ ka units digit 0” → kabhi nahi.

WHY yeh tareeka? “Hamesha” ke liye wajah chahiye; “kabhi nahi”/“kabhi-kabhi” ke liye ek counter-example kaafi.

Step 4: Hathiyaar 4: Ginti (codes aur combinations)

4-ank ka PIN (0–9): har jagah 10 choice → 10⁴ = 10,000. 5 ank → 10⁵.

WHY? Har nayi jagah pichle saare options ke saath judti hai: × 10 (Post 4 — Combinations Aur Powers).

🪜 Atak Gaye? Aise Socho

Pehle dono powers ke base factorise karo: 4 = 2², 8 = 2³, 27 = 3³.
Same base ban gaya? Exponent compare karo. Nahi? Exponents ka HCF lo aur same exponent banao.
Units digit ke liye pehle 4–5 powers likh ke chakra dhundho.
Exponent ko chakra ki lambaai se bhaag do; shesh (remainder) = position (shesh 0 = aakhri).
“Hamesha” bolne se pehle 0, 1, negative aur fraction pe test karo.

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🔐 Deep Concept: Ek Cheez Barabar Karo: Tulna Ka Raaz

Do cheezein jinki do-do “taakat” alag hon, unki seedhi tulna mushkil. Jaise: 3 kg ₹150 wala aam vs 2 kg ₹110 wala. Hum kya karte hain? 1 kg pe laate hain (ek cheez barabar)!

Powers mein bhi yahi: base aur exponent dono alag → kuch nahi pata. Ek ko barabar karo (same base ya same exponent), phir faisla saaf.

Isliye exponent ke niyam itne kaam ke: (a^m)ⁿ = a^mn se 2³⁰ ko 8¹⁰ bana sake. Niyam = roop badalne ka auzaar, number wahi.

Base < 1 ho to ulta! (1/2)³ = 1/8 < (1/2)² = 1/4. Bada exponent = chhota number. Isliye “hamesha” bolne se pehle fractions check karo.

Unifying idea: ek cheez barabar karo, phir doosri compare karo. Yahi ratio, fractions aur percentage ki tulna mein bhi hota hai. 🎯

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✏️ Solved Examples

Example 1 (Easy): Kaun bada: 5¹² ya 11⁸?

Exponents 12, 8 ka HCF 4. 5¹² = (5³)⁴ = 125⁴. 11⁸ = (11²)⁴ = 121⁴.

125 > 121 → 5¹² bada.

Check: 5¹² = 24,41,40,625; 11⁸ = 21,43,58,881 ✅

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Example 2 (Medium): 3⁵⁰ ka units digit?

Chakra: 3, 9, 7, 1 (lambaai 4). 50 = 4 × 12 + 2 → 2nd position → 9.

Check: 3² = 9, 3⁶ = 729 (6 = 4 + 2), dono 9 pe khatam ✅

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Example 3 (NCERT style): Hamesha, kabhi-kabhi ya kabhi nahi: “n² > n”

n = 3: 9 > 3 ✅. n = 1: 1 = 1 ❌. n = 1/2: 1/4 < 1/2 ❌.

→ Kabhi-kabhi (n > 1 ya n < 0 pe sach).

Check: n = −2: 4 > −2 ✅, to negative pe bhi sach ✅

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Meenu ki chunauti: 2³⁰ = 8¹⁰, 3²⁰ = 9¹⁰ → 3²⁰ bada! (Asal: 2³⁰ ≈ 107 crore, 3²⁰ ≈ 349 crore, 3 guna se zyada.) Golu ka “bada exponent jeetega” galat nikla. 😅

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🏠 Real Life Mein Yeh Hathiyaar

🔐 Password: 6-ank PIN = 10⁶ = 10 lakh options; letters bhi ho (36 choice) to 36⁶ ≈ 2 arab! Isliye mixed password zyada surakshit.
🚗 Number plate: “MH 12 AB 1234” jaise patterns: letters 26² × ank 10⁴ = 67.6 lakh plates har zile ke liye.
📅 Hafte ka din: din ka chakra 7 ka. Aaj ravivaar, to 100 din baad? 100 = 7 × 14 + 2 → mangalvaar. Units digit jaisa hi chakra!
🛒 Offer compare: “2 ka 3 guna” vs “3 ka 2 guna” jaisi tulna: ek cheez barabar karo.
💾 Memory: 1 GB = 2³⁰ byte ≈ 10⁹; isliye phone ke “64 GB” mein thoda kam dikhta hai.

Chakra, ginti, tulna: teeno roz ke kaam mein chhupe hain.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Base ek doosre ki power (2, 4, 8)	✅ Same base banao	Exponent dekho
Exponents ka HCF (30, 20)	✅ Same exponent banao	Base dekho
Bahut badi power ka aakhri ank	✅ Units digit chakra	Shesh = position
“Hamesha” daava	✅ Counter-example dhundho	0, 1, fraction
Options ki ginti	✅ Choices ka guna	10 × 10 × 10…

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❌→✅ Worked Mistake

Sawaal: 2⁴⁰ aur 5²⁰ mein kaun bada?

❌ Galat: “Base 5 bada → 5²⁰” ya “exponent 40 bada → 2⁴⁰” — sirf ek taakat dekhi.

✅ Sahi: 2⁴⁰ = (2²)²⁰ = 4²⁰. Ab 4²⁰ vs 5²⁰ → 5²⁰ bada. Check: 2⁴⁰ ≈ 1.1 × 10¹², 5²⁰ ≈ 9.5 × 10¹³ ✅

Doosri galti (units digit): 7⁴⁴ ka units digit: 44 ÷ 4 shesh 0 → “0th position”? Nahi! Shesh 0 = aakhri position → 1.

🧠 Rule of thumb: Pehle ek cheez barabar karo (base ya exponent), phir doosri compare karo. Shesh 0 = chakra ka aakhri ank.

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📝 10 MCQs

Q1. Kaun bada: 2¹⁰⁰ ya 4⁴⁹?

(a) 2¹⁰⁰

(b) 4⁴⁹

(c) Barabar

(d) Pata nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: 4⁴⁹ = 2⁹⁸, aur 2¹⁰⁰ = 4 × 2⁹⁸ bada ✅
(b) galat: 4⁴⁹ = 2⁹⁸ hai, jo 2¹⁰⁰ se chhota.
(c) galat: 2¹⁰⁰ ≠ 2⁹⁸, barabar nahi hain.
(d) galat: Same base banane se turant pata chalta hai.

Q2. 2⁴³ ka units digit?

(a) 2

(b) 4

(c) 8

(d) 6

Solution dekho

Sahi jawab: (c)

(a) galat: 43 ÷ 4 shesh 3, pehli position nahi.
(b) galat: Shesh 3 hai, 2nd position nahi.
(c) sahi: Chakra 2, 4, 8, 6 → shesh 3 → 8 ✅
(d) galat: 6 aakhri position (shesh 0) pe aata.

Q3. “a⁰ = 1” (a ≠ 0) hai…

(a) Hamesha

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf a = 1

Solution dekho

Sahi jawab: (a)

(a) sahi: Har non-zero a ke liye sach ✅
(b) galat: Koi counter-example nahi hai yahan.
(c) galat: 5⁰ = 1, to “kabhi nahi” galat.
(d) galat: 2⁰, 7⁰ sab 1 hain, sirf 1 nahi.

Q4. 3-ank ka code (har ank 0–9): kitne options?

(a) 30

(b) 100

(c) 999

(d) 1,000

Solution dekho

Sahi jawab: (d)

(a) galat: 3 × 10 joda; guna karna tha.
(b) galat: Yeh 2-ank ke codes ki ginti hai.
(c) galat: 000 bhi ek code hai, use bhi gino.
(d) sahi: 10 × 10 × 10 = 1,000 ✅

Q5. Kaun bada: 3⁴⁰ ya 4³⁰?

(a) 3⁴⁰

(b) 4³⁰

(c) Barabar

(d) Pata nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: (3⁴)¹⁰ = 81¹⁰ > (4³)¹⁰ = 64¹⁰ ✅
(b) galat: 64¹⁰ chhota hai 81¹⁰ se.
(c) galat: 81 aur 64 barabar nahi hain.
(d) galat: HCF 10 lekar same exponent banao.

Q6. (1/2)⁵ aur (1/2)³ mein kaun bada?

(a) (1/2)⁵

(b) (1/2)³

(c) Barabar

(d) Dono 0

Solution dekho

Sahi jawab: (b)

(a) galat: 1/32 chhota hai 1/8 se.
(b) sahi: 1/8 > 1/32: base < 1 pe bada exponent chhota ✅
(c) galat: 1/8 aur 1/32 barabar nahi.
(d) galat: Powers kabhi 0 nahi banti yahan.

Q7. 9¹⁵ ka units digit?

(a) 9

(b) 1

(c) 3

(d) 7

Solution dekho

Sahi jawab: (a)

(a) sahi: Chakra 9, 1; 15 odd → 9 ✅
(b) galat: 1 sirf even power pe aata hai.
(c) galat: 3 to 3 ki powers mein aata hai.
(d) galat: 7 bhi 3/7 ke chakra ka ank hai.

Q8. “2ⁿ + 1 prime hai” hai…

(a) Hamesha

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf n = 0

Solution dekho

Sahi jawab: (b)

(a) galat: n = 3: 9 prime nahi, to hamesha nahi.
(b) sahi: n = 1: 3 prime, n = 3: 9 nahi → kabhi-kabhi ✅
(c) galat: n = 2: 5 prime, to “kabhi nahi” galat.
(d) galat: n = 1, 2, 4 pe bhi prime milta hai.

Q9. 6¹⁰⁰ ka units digit?

(a) 6

(b) 0

(c) 4

(d) 2

Solution dekho

Sahi jawab: (a)

(a) sahi: 6 × 6 = 36: hamesha 6 ✅
(b) galat: 6 ki koi power 0 pe khatam nahi hoti.
(c) galat: 4 nahi aata: 6 × 6 = 36.
(d) galat: 2 nahi aata: hamesha 6 rehta.

Q10. 2²⁰ aur 10⁶ mein kaun bada?

(a) 2²⁰

(b) 10⁶

(c) Barabar

(d) Pata nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: 2²⁰ = 10,48,576 > 10,00,000 ✅
(b) galat: 10⁶ thoda chhota hai 2²⁰ se.
(c) galat: 1,048,576 ≠ 1,000,000.
(d) galat: 2¹⁰ ≈ 1,024 se turant pata chalta.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Same base banana kab mumkin nahi?

Jab base alag primes hon (2 aur 3). Tab same exponent try karo, ya andaaza (2¹⁰ ≈ 10³).

2. Units digit chakra hamesha 4 ka hota hai?

Nahi: 2, 3, 7, 8 ka 4; 4, 9 ka 2; 0, 1, 5, 6 ka 1. Par 4 se hamesha kaam chal jaata hai.

3. Ek counter-example se daava galat kaise?

“Hamesha” ka matlab har case. Ek bhi case fail → hamesha galat. Isliye counter-example itna taakatwar hai.

4. Kya 0⁰ = 1?

Yeh vivaadit hai; school maths mein ise “paribhaashit nahi” maana jaata hai. Sawaal mein a ≠ 0 likha hota hai isi liye.

5. Olympiad mein aise sawaal aate hain?

Haan, bahut! Units digit, compare powers, last two digits: sab isi soch se.

6. Last do ank kaise nikaalein?

Wahi idea: sirf aakhri do ank pe guna karte jao, aur chakra dhundho. Jaise 7⁴ = 2401 → last do ank 01, to har 4th power …01!

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy Kaun bada: 2¹² ya 4⁵?

Solution dekho

4⁵ = 2¹⁰ → 2¹² bada. Check: 4,096 > 1,024 ✅

Q2. Easy 4¹⁷ ka units digit?

Solution dekho

Chakra 4, 6; 17 odd → 4. Check: 4³ = 64 ✅

Q3. Easy 5-ank ke PIN kitne?

Solution dekho

10⁵ = 1,00,000. WHY: har ank ke 10 choice, guna.

Q4. Easy “Kisi bhi power of 10 ka units digit 0” (exponent ≥ 1)?

Solution dekho

Hamesha. WHY: 10 se guna karne pe aakhri ank 0.

Q5. Medium Kaun bada: 3³⁰ ya 5²⁰?

Solution dekho

27¹⁰ vs 25¹⁰ → 3³⁰ bada. Check: 27 > 25 ✅

Q6. Medium 8²⁵ ka units digit?

Solution dekho

Chakra 8, 4, 2, 6; 25 = 24 + 1 → 8. Check: 8⁵ = 32,768 ✅

Q7. Medium “a² ka units digit 2, 3, 7 ya 8 ho sakta hai?”

Solution dekho

Kabhi nahi. Squares ke units: 0, 1, 4, 5, 6, 9. Check: 0–9 ke squares gino ✅

Q8. Medium 2 letter (A–Z) + 3 ank wale codes kitne?

Solution dekho

26² × 10³ = 676 × 1,000 = 6,76,000. Check: 26 × 26 = 676 ✅

Q9. NCERT style Kaun bada: 2⁵⁰, 3³⁰, 5²⁰?

Solution dekho

HCF 10 se same exponent: (2⁵)¹⁰ = 32¹⁰, (3³)¹⁰ = 27¹⁰, (5²)¹⁰ = 25¹⁰ → 2⁵⁰ > 3³⁰ > 5²⁰. Check: 32 > 27 > 25 ✅

Q10. NCERT style 7²⁰²⁶ ka units digit?

Solution dekho

2026 = 4 × 506 + 2 → 2nd position → 9. Check: 7² = 49 ✅

Q11. NCERT style 2¹⁰ + 3¹⁰ ka units digit?

Solution dekho

2¹⁰: 10 = 8 + 2 → 4. 3¹⁰ → 9. 4 + 9 = 13 → 3. Check: 1,024 + 59,049 = 60,073 ✅

Q12. NCERT style “2ⁿ > n²” — hamesha, kabhi-kabhi, kabhi nahi? (n = 1, 2, 3…)

Solution dekho

n = 1: 2 > 1 ✅; n = 2: 4 = 4 ❌; n = 3: 8 < 9 ❌; n = 5: 32 > 25 ✅ → kabhi-kabhi (n = 1 aur n ≥ 5). WHY: lambi race mein exponential jeetta (Post 7 — Linear Vs Exponential Growth).

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Kaun bada: 16⁸ ya 8¹¹?

Solution dekho

2³² vs 2³³ → 8¹¹ bada (do guna). Check: 33 − 32 = 1 → 2 guna ✅

Q2. 3²⁰²⁶ − 1 ka units digit?

Solution dekho

2026 shesh 2 → 9. 9 − 1 = 8. WHY: chakra 3, 9, 7, 1.

Q3. Kya kisi bhi n ke liye 4ⁿ ka units digit 0 ho sakta hai?

Solution dekho

Nahi. 0 ke liye 2 aur 5 dono factor chahiye, 4ⁿ mein 5 kabhi nahi (prime factorisation, Post 2 — Exponent Aur Base Kya Hain?).

Q4. Kitne 4-ank ke PIN mein kam se kam ek 7 hai?

Solution dekho

Complement: kul 10⁴ − bina 7 wale 9⁴ = 10,000 − 6,561 = 3,439. WHY: “kam se kam ek” = sab − koi nahi.

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🏆 Olympiad Corner

Q1. 2¹⁰⁰ ke kitne ank (digits) honge? (2¹⁰ ≈ 1,024)

Solution dekho

2¹⁰⁰ = (1,024)¹⁰ ≈ 1.27 × 10³⁰ → 31 ank.

Q2. 1¹ + 2² + 3³ + 4⁴ + 5⁵ ka units digit?

Solution dekho

1 + 4 + 7 + 6 + 5 = 23 → 3.

Q3. Kaun bada: 99²⁰ ya 9999¹⁰?

Solution dekho

99²⁰ = 9,801¹⁰ < 9,999¹⁰ → 9999¹⁰ bada.

Q4. 17¹⁷ ka units digit?

Solution dekho

7 ka chakra; 17 = 16 + 1 → 7.

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💗 CharuMam Motivation

Golu ne sirf ek taakat dekhi, Meenu ne ek cheez barabar karke dono dekhi. Yahi “tricky” sawaal ko aasaan banaata hai. 🧠

Poora Power Play chapter khatam! Aapne exponential growth se lekar arab second tak sab samjha. Ab koi bhi badi power aapko daraa nahi sakti! 🚀

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🟢 Master Recap

💡 Chaabi: Powers ke tricky sawaal mein pehle same base ya same exponent banao. Jab ek cheez barabar ho, to doosri dekh ke faisla turant. Exponent ke niyam isi ke auzaar hain.
Compare: same base ya same exponent banao.
Base > 1: bada exponent = bada number; base < 1: ulta.
Units digit ka chakra: shesh = position, shesh 0 = aakhri.
“Hamesha” ke liye wajah, “kabhi nahi/kabhi-kabhi” ke liye counter-example.
Codes: har jagah ke choices ka guna.
“Kam se kam ek” = kul − koi nahi.
2³⁰ < 3²⁰: Meenu jeeti!

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👉 Aage Kya?

Chapter 2 poora! Ab chalte hain numbers ki kahani mein: insaan ne ginti kaise shuru ki, Roman numerals, alag-alag base aur zero ka jaadu. Chapter 3: A Story of Numbers ka hub dekhiye. 📜

← Pichla: Post 8 — Bade Numbers Ka Andaaza 📚 Chapter 2 ke saare posts Agla chapter: Ch 3 — A Story of Numbers →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Meenu Ki Chunauti: 230 Ya 320?

📖 Concept: Tricky Sawaal Ke 4 Hathiyaar

Step 1: Hathiyaar 1: Compare karna

Step 2: Hathiyaar 2: Units digit ka chakra

Step 3: Hathiyaar 3: Hamesha / Kabhi-kabhi / Kabhi nahi

Step 4: Hathiyaar 4: Ginti (codes aur combinations)

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Ek Cheez Barabar Karo: Tulna Ka Raaz

✏️ Solved Examples

🏠 Real Life Mein Yeh Hathiyaar

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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