Exponent Ke Niyam: Guna Aur Power | Class 8 Maths Book 1 Ch 2

📘 Class 8 · Ganita Prakash Book 1Ch 2 · Power PlayPost 3 / 9⏱ 25–30 min

Exponent Ke Niyam: Guna Aur Power ⚙️

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 = ?
5³ mein base aur exponent?
3² × 3³ ko lamba likho.

Answers dekho

32 (2 → 4 → 8 → 16 → 32)
Base 5 (jo guna ho raha), exponent 3 (kitni baar)
(3 × 3) × (3 × 3 × 3): kul paanch 3

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 2 — Exponent Aur Base Kya Hain? dobara dekh lijiye.

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🎬 Kahani: School Library Ki Ginti

School library mein Golu ki duty lagi: “Ginti karo kitni kitaabein hain.” Librarian ma’am ne bataaya:

“5 almaariyan, har almaari mein 5 khaane, har khaane mein 5 dabbe, har dabbe mein 5 kitaabein.”

Golu: “5 × 5 = 25 khaane, 25 × 5 = 125 dabbe, 125 × 5 = 625 kitaabein.” Yaani $5^4$.

Tabhi ma’am boli: “Aur har kitaab mein 5 chapter, har chapter mein 5 chitra, har chitra mein 5 rang. Kul kitne rang?” Golu ne sar pakad liya. 😵 Kya phir shuru se guna? Post ke end tak Golu ek line mein jawab dega. 🚀

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📖 Concept: Teen Golden Niyam (Sab Ginti Se)

💡 Is post ki ek badi baat: Exponent ke saare niyam ek hi sawaal se nikalte hain: “kul kitni baar guna hua?” Lamba likh ke ginti karo, niyam khud ban jaayega. Ratna nahi, ginna hai.

Step 1: Niyam 1: same base ka guna → powers jodo

Kitaabein $5^4$. Aage 3 level aur, har level × 5: $5^4 \times 5^3$.

$5^4 \times 5^3 = (5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5) = 5^7$

$n^a \times n^b = n^{a+b}$

WHY jodna? Pehle group mein a paanch, doosre mein b paanch. Bracket hata do: ek line mein kul a + b paanch. Ginti jud gayi, guna nahi hui!

Golu ka jawab: $5^7$ = 625 × 125 = 78,125 rang.

Step 2: Niyam 2: power ki power → powers guna

$(2^3)^4$ = $2^3$ ka block 4 baar: $2^3 \times 2^3 \times 2^3 \times 2^3 = 2^{3+3+3+3} = 2^{12}$.

$(n^a)^b = n^{a \times b} = (n^b)^a$

WHY guna? a wala block b baar aaya: a + a + … (b baar) = a × b. Baar-baar jodna = guna. Niyam 2 asal mein Niyam 1 hi hai, baar-baar laga hua!

Isliye $2^{12}$ = $(2^3)^4$ = $(2^4)^3$ = $(2^6)^2$ = $(2^2)^6$: 12 ko kisi bhi do numbers ke guna mein todo.

Step 3: Niyam 3: same exponent, alag base → base guna

$2^3 \times 5^3 = (2 \times 2 \times 2) \times (5 \times 5 \times 5)$. Jodiyaan banao: $(2 \times 5)(2 \times 5)(2 \times 5) = 10^3 = 1000$.

$m^a \times n^a = (m \times n)^a, \qquad \dfrac{m^a}{n^a} = \left(\dfrac{m}{n}\right)^a$

WHY? Dono taraf barabar ginti (a-a) hai, isliye har m ke saath ek n ki jodi banti hai. Guna mein order badal sakte ho, to jodiyaan bana lo.

Jaise $\dfrac{12^3}{4^3} = 3^3 = 27$ (check: 1,728 ÷ 64 = 27 ✅).

Step 4: Ek sawaal, do tareeke: kab kaunsa?

Sawaal: $4^5 \times 25^5$ = ?

Tareeka A (alag-alag): $4^5 = 1024$, $25^5 = 97{,}65{,}625$, phir guna. Lamba aur galti ka khatra!

Tareeka B (Niyam 3): $(4 \times 25)^5 = 100^5 = 10^{10}$. Ek line! ✅

Guidance: exponent same aur base ka guna “sundar” (10, 100) bane → Niyam 3. Base same → Niyam 1. Bracket ke bahar power → Niyam 2. Kuch same nahi → seedha calculate.

Step 5: Kab niyam NAHI lagta

$2^3 \times 3^2$: base bhi alag, exponent bhi. Koi chhota form nahi: 8 × 9 = 72.
$2^3 + 2^4$: jod pe koi niyam nahi! 8 + 16 = 24, jabki $2^7 = 128$.

WHY? Saare niyam “kitni baar guna” ki ginti se bane hain. Jod mein guna ki ginti hoti hi nahi, to niyam ka base hi nahi.

🪜 Atak Gaye? Aise Socho

Niyam bhool gaye? Chhote numbers pe lamba likho: 2² × 2³ = (2 × 2)(2 × 2 × 2) → paanch 2 → 2⁵. Niyam wapas mil gaya!
Pehle dekho: base same hai ya exponent same? Wahi batata hai kaunsa niyam.
Bracket ke bahar power ho to “block kitni baar” socho → guna.
Akela number = power 1 (7 = 7¹). Ise jodna mat bhoolna.
Jawab ko chhote case pe value nikaal ke check karo.

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🔐 Deep Concept: Guna Ko Jod Mein Badalna: Exponents Ka Jaadu

2 ki powers ki ek table banao:

Power	1	2	3	4	5	6	7	8	9	10
Value	2	4	8	16	32	64	128	256	512	1024

Ab 16 × 64 = ? Table mein dekho: 16 = power 4, 64 = power 6. 4 + 6 = 10 → 1,024! Guna kiya hi nahi, sirf jod kiya. 🤯

WHY? Yahi Niyam 1: $2^4 \times 2^6 = 2^{10}$. Jab sab numbers ek hi base ki powers mein likhe hon, to bada guna = chhota jod.

Itihaas: 400 saal pehle scientists ne isi idea se “logarithm tables” banaayi, taaki grahon ki doori jaise bade guna jod se ho jaayein. Calculator se pehle yahi sabse bada “calculator” tha!

Unifying idea: powers ki duniya mein guna → jod, power ki power → guna, bhaag → ghataav (Post 5 — Zero Aur Negative Exponent). Har operation ek seedhi neeche utar jaata hai. Isliye exponents itne kaam ke hain. 🎯

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✏️ Solved Examples

Example 1 (Easy): Ek power mein likho aur value nikaalo: 3⁵ × 3²

Same base → powers jodo: $3^{5+2} = 3^7$ = 2,187.

Check: 3⁵ = 243, 3² = 9, 243 × 9 = 2,187 ✅

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Example 2 (Medium): Simplify: (5²)³ × 5

(5²)³ = 5⁶ (Niyam 2). Akela 5 = 5¹. 5⁶ × 5¹ = 5⁷ = 78,125.

Check: 25³ = 15,625, × 5 = 78,125 ✅

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Example 3 (NCERT style): Value nikaalo: 2⁶ × 5⁶

Same exponent → (2 × 5)⁶ = 10⁶ = 10,00,000.

Check: 64 × 15,625 = 10,00,000 ✅ (64 × 15,625 = 64 × 15,000 + 64 × 625 = 9,60,000 + 40,000)

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Golu ka jawab: “Kitaabein 5⁴, aur 3 level aur: 5⁴ × 5³ = 5⁷ = 78,125 rang!” Ma’am ne kaha: “10 second mein? Shabash!” 📚

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🏠 Real Life Mein Exponent Niyam

📦 Packing: dabbe ke andar dabbe: har level ka guna, isliye levels ki powers jodo.
💾 Memory: 1 GB = 2¹⁰ MB aur 1 MB = 2¹⁰ KB → 1 GB = 2²⁰ KB (Niyam 1).
📐 Units badalna: 1 m = 10² cm, to 1 m³ = (10²)³ = 10⁶ cm³ (Niyam 2). Paani ki tanki ka hisaab!
🧮 Mental maths: 25³ × 4³ = 100³ = 10 lakh (Niyam 3).
🔊 Awaaz (decibel): har 10 dB = 10 guna taaqat; 20 dB = 10 × 10 = 10² guna. Powers jud rahi hain!

Jahan bhi “guna ke andar guna” ho, wahan powers jodne ya guna karne ka mauka hai.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Same base, guna (n^a × n^b)	✅ Powers jodo	Ginti jud jaati hai
Power ki power ((n^a)^b)	✅ Powers guna	Block b baar
Same exponent, alag base (m^a × n^a)	✅ Base guna, exponent wahi	Jodiyaan
Jod (n^a + n^b)	❌ Koi niyam nahi	Pehle value nikaalo
Base bhi alag, exponent bhi alag	❌ Chhota form nahi	Seedha calculate

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❌→✅ Worked Mistake

Sawaal: 2³ × 2⁴ = ?

❌ Galat 1: “Powers guna karo: 2¹².” ❌ Galat 2: “Base bhi guna karo: 4⁷.”

Galti kya hai? Guna ke saath powers jodte hain (guna sirf power-ki-power mein). Aur base sirf “kaun guna ho raha” batata hai, woh khud guna nahi hota.

✅ Sahi: 2³⁺⁴ = 2⁷ = 128. Check: 8 × 16 = 128 ✅, jabki 2¹² = 4,096 aur 4⁷ = 16,384, dono bahut bade.

Doosri common galti: 3² + 3² = 3⁴. Nahi! 9 + 9 = 18 = 2 × 3². Jod pe niyam nahi.

🧠 Rule of thumb: Shak ho to chhota example lamba likho aur gino. Ginti kabhi jhooth nahi bolti!

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📝 10 MCQs

Q1. 7² × 7⁵ = ?

(a) 7¹⁰

(b) 7⁷

(c) 49⁷

(d) 14⁷

Solution dekho

Sahi jawab: (b)

(a) galat: Powers guna kar diye; guna sirf power-ki-power mein hota hai.
(b) sahi: Do 7 + paanch 7 = saat 7 → 7⁷ ✅
(c) galat: Base bhi guna kar diya; base wahi rehta hai.
(d) galat: Base jod diye (7 + 7); base nahi badalta.

Q2. (3²)⁴ = ?

(a) 3⁶

(b) 3⁸

(c) 9⁸

(d) 3¹⁶

Solution dekho

Sahi jawab: (b)

(a) galat: 2 + 4 jod diya; block 4 baar hai, to guna chahiye.
(b) sahi: 3² ka block 4 baar: 2 × 4 = 8 → 3⁸ ✅
(c) galat: Base ko bhi square kar diya, double counting.
(d) galat: Exponent ka square (4²) kar diya, galat.

Q3. 2⁴ × 5⁴ = ?

(a) 10⁸

(b) 7⁴

(c) 10⁴

(d) 10¹⁶

Solution dekho

Sahi jawab: (c)

(a) galat: Exponent jod diye; yahan jodi bani hain, exponent wahi.
(b) galat: Base jod diye; Niyam 3 mein base guna hote hain.
(c) sahi: Chaar (2 × 5) jodiyaan → 10⁴ ✅
(d) galat: Exponent guna kar diye, koi tark nahi.

Q4. 2³ + 2³ = ?

(a) 2⁶

(b) 4³

(c) 2⁴

(d) 2⁹

Solution dekho

Sahi jawab: (c)

(a) galat: Jod pe guna wala niyam laga diya.
(b) galat: Base jod diye; jod pe aisa niyam nahi.
(c) sahi: 8 + 8 = 16 = 2⁴ ✅ (do baar 2³ = 2 × 2³)
(d) galat: Exponent guna kiye, bilkul galat.

Q5. a⁵ × a × a² = ?

(a) a⁷

(b) a⁸

(c) a¹⁰

(d) 3a⁸

Solution dekho

Sahi jawab: (b)

(a) galat: Beech wala akela a (power 1) bhool gaye.
(b) sahi: Powers 5 + 1 + 2 = 8 → a⁸ ✅
(c) galat: Powers guna kar diye, galat niyam.
(d) galat: Coefficient 3 kahan se? Teen terms ka guna hai, jod nahi.

Q6. 12³ ÷ 3³ = ?

(a) 4³

(b) 4⁰

(c) 9³

(d) 36³

Solution dekho

Sahi jawab: (a)

(a) sahi: Same exponent: (12 ÷ 3)³ = 4³ = 64 ✅
(b) galat: Exponent ghata diye; woh tab jab base same ho.
(c) galat: Base ghata diye (12 − 3); bhaag hai, ghataav nahi.
(d) galat: Base guna kar diye; yahan bhaag hai.

Q7. Kaun sa 2¹² ke barabar NAHI?

(a) (2³)⁴

(b) 4⁶

(c) 8⁴

(d) (2⁶)⁶

Solution dekho

Sahi jawab: (d)

(a) galat: 3 × 4 = 12, to barabar hai.
(b) galat: 4 = 2², to 4⁶ = 2¹², barabar.
(c) galat: 8 = 2³, to 8⁴ = 2¹², barabar.
(d) sahi: 6 × 6 = 36 → yeh 2³⁶ hai, barabar nahi ✅

Q8. (−2)³ × (−2)² = ?

(a) 32

(b) −32

(c) −64

(d) 64

Solution dekho

Sahi jawab: (b)

(a) galat: Sign bhool gaye: kul 5 minus hain, odd.
(b) sahi: Kul (−2)⁵: odd power → −32 ✅
(c) galat: Powers guna kar ke 2⁶ jaisa socha.
(d) galat: Sign aur power dono galat.

Q9. x²y³ × x⁴y = ?

(a) x⁶y⁴

(b) x⁸y³

(c) x⁶y³

(d) (xy)¹⁰

Solution dekho

Sahi jawab: (a)

(a) sahi: x ki 2 + 4, y ki 3 + 1 → x⁶y⁴ ✅
(b) galat: x ki powers guna ki aur y ka akela 1 bhool gaye.
(c) galat: y ke saath wala akela y (power 1) bhool gaye.
(d) galat: Alag letters ki powers aapas mein nahi judti.

Q10. 25² × 4² = ?

(a) 100⁴

(b) 10⁴

(c) 29²

(d) 20⁴

Solution dekho

Sahi jawab: (b)

(a) galat: Exponent jod diye; Niyam 3 mein exponent wahi rehta.
(b) sahi: Jodi bana ke (25 × 4)² = 100² = 10⁴ ✅
(c) galat: Base jod diye (25 + 4); guna chahiye.
(d) galat: Galat jodi; 20⁴ = 1,60,000 hai, 10,000 nahi.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Niyam 1 mein jod, Niyam 2 mein guna. Confusion kaise hataayein?

Socho “kitne base guna hue”. n^a × n^b: pehle a, phir b → a + b. (n^a)^b: a wala block b baar → a × b.

2. Kya niyam negative ya fraction base pe bhi chalte hain?

Haan. Ginti base pe nirbhar nahi: (−3)² × (−3)³ = (−3)⁵, (1/2)² × (1/2)³ = (1/2)⁵.

3. m^a × n^b (sab alag) ka kya?

Koi chhota form nahi. Value nikaalo. Kabhi-kabhi base ko tod ke same bana sakte ho: 4³ × 2⁵ = 2⁶ × 2⁵ = 2¹¹.

4. (ab)ⁿ = aⁿbⁿ hota hai?

Haan, yahi Niyam 3 ulta: (2 × 3)² = 36 = 4 × 9 ✅

5. (a + b)² = a² + b²?

Nahi! (2 + 3)² = 25, par 4 + 9 = 13. Power jod pe nahi baant-ti. Sahi formula Chapter 6 mein: Book 1 Ch 6 — (a + b)² Ka Formula.

6. Power ka power ka power?

((2²)³)² = 2^{2 × 3 × 2} = 2¹². Bas guna karte jao.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy Ek power mein: 4³ × 4⁵

Solution dekho

4⁸. WHY: teen 4 + paanch 4 = aath 4.

Q2. Easy Ek power mein: (6²)⁵

Solution dekho

6¹⁰. WHY: 6² ka block 5 baar → 2 × 5.

Q3. Easy Value: 5³ × 2³

Solution dekho

10³ = 1,000. Check: 125 × 8 = 1,000 ✅

Q4. Easy Ek power mein: y × y⁴ × y³

Solution dekho

1 + 4 + 3 = y⁸. Check: akela y = y¹ gina ✅

Q5. Medium Value: 2⁵ × 2³ (pehle ek power mein)

Solution dekho

2⁸ = 256. Check: 32 × 8 = 256 ✅

Q6. Medium 3¹² ko power-ki-power mein do tareeke se likho.

Solution dekho

(3⁴)³, (3⁶)², (3²)⁶, (3³)⁴… WHY: 12 ko do numbers ke guna mein todo.

Q7. Medium Value: 20³ ÷ 5³

Solution dekho

(20 ÷ 5)³ = 4³ = 64. Check: 8,000 ÷ 125 = 64 ✅

Q8. Medium Simplify: a³b² × a²b⁵

Solution dekho

a⁵b⁷. Check: a = 1, b = 2 rakh ke: 4 × 32 = 128 = 2⁷ ✅

Q9. NCERT style Ek bagiche mein phool har hafte double hote hain. 1 phool se shuru; 3 hafte baad sab ek naye bagiche mein jahan har hafte 5 guna. Aur 3 hafte baad kitne?

Solution dekho

1 × 2³ × 5³ = 10³ = 1,000. Check: 8 phool, phir 8 → 40 → 200 → 1,000 ✅

Q10. NCERT style Value: 50⁴ × 2⁴

Solution dekho

(50 × 2)⁴ = 100⁴ = 10⁸ = 10 crore. Check: 100⁴ mein 4 × 2 = 8 zero ✅

Q11. NCERT style 9⁵ ko 3 ki power mein likho.

Solution dekho

9 = 3² → (3²)⁵ = 3¹⁰. Check: 9⁵ = 59,049 = 3¹⁰ ✅

Q12. NCERT style Kya 8³ = 2⁹? Dikhao.

Solution dekho

8 = 2³, (2³)³ = 2⁹. Check: 512 = 512 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. 2^x × 2³ = 2¹⁰. x?

Solution dekho

x + 3 = 10 → x = 7. WHY: same base, to powers ka jod barabar hona chahiye.

Q2. 4⁵ aur 2⁹ mein bada kaun? Bina poora calculate kiye.

Solution dekho

Base same karo: 4⁵ = (2²)⁵ = 2¹⁰ > 2⁹. 4⁵, theek do guna. Trick: compare se pehle base same karo.

Q3. (2³)² aur 2^(3²) same hain?

Solution dekho

(2³)² = 2⁶ = 64. 2⁹ = 512. Nahi. Bracket batata hai pehle kya karna hai.

Q4. 125⁴ × 8⁴ kitne digit ka number hai?

Solution dekho

(125 × 8)⁴ = 1,000⁴ = 10¹² → 1 ke baad 12 zero → 13 digit.

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🏆 Olympiad Corner

Q1. 3^a × 9² = 3¹⁰. a?

Solution dekho

9² = 3⁴. a + 4 = 10 → a = 6.

Q2. 2¹⁰ × 5⁸ kitne zero pe khatam hota hai?

Solution dekho

= 2² × (2 × 5)⁸ = 4 × 10⁸ → 8 zero. WHY: har zero ek (2 × 5) jodi se banta hai, aur 5 sirf 8 hain.

Q3. 4,096 (= 2¹²) ko a^b ke roop mein kitne tareeke se likh sakte ho (a, b counting numbers, b > 1)?

Solution dekho

b, 12 ka factor: b = 2, 3, 4, 6, 12 → 64², 16³, 8⁴, 4⁶, 2¹². 5 tareeke.

Q4. Kaun sa bada: 2³⁰, 3²⁰, 5¹⁰?

Solution dekho

Exponent same karo: 2³⁰ = 8¹⁰, 3²⁰ = 9¹⁰, 5¹⁰. Ab base compare: 3²⁰ sabse bada.

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💗 CharuMam Motivation

Golu ne library ki ginti 10 second mein kar di, kyunki usne niyam ka kyun samjha tha, ratta nahi maara. Kal agar niyam bhool bhi jaaye, to 30 second mein lamba likh ke wapas bana lega. Yahi asli taaqat hai! 🧠

Aap bhi har formula ke peeche ka “kyun” dhoondho. Jo samajh mein aata hai, woh kabhi bhoolta nahi. 🤝

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🟢 Master Recap

💡 Chaabi: Exponent ke saare niyam ek hi sawaal se nikalte hain: “kul kitni baar guna hua?” Lamba likh ke ginti karo, niyam khud ban jaayega. Ratna nahi, ginna hai.
n^a × n^b = n^a+b (same base, powers jodo).
(n^a)^b = n^ab (block b baar → guna).
m^a × n^a = (mn)^a; m^a ÷ n^a = (m ÷ n)^a.
Jod pe koi niyam nahi: 2³ + 2⁴ ≠ 2⁷.
Powers ki duniya mein guna → jod (log tables ka idea).
Compare karne se pehle base ya exponent same karo.
Shak ho to lamba likh ke gino.

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👉 Aage Kya?

Ab ek mazedaar sawaal: 4 shirt aur 3 topi se kitne outfit? 4 digit ke lock mein kitne password? Jawab mein phir powers aayengi! Agle post mein: combinations aur powers. 🔐

← Pichla: Post 2 — Exponent Aur Base Kya Hain?📚 Chapter 2 ke saare posts Agla: Post 4 — Combinations Aur Powers →

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

✅ Pehle Yeh Aata Hai?

🎬 Kahani: School Library Ki Ginti

📖 Concept: Teen Golden Niyam (Sab Ginti Se)

Step 1: Niyam 1: same base ka guna → powers jodo

Step 2: Niyam 2: power ki power → powers guna

Step 3: Niyam 3: same exponent, alag base → base guna

Step 4: Ek sawaal, do tareeke: kab kaunsa?

Step 5: Kab niyam NAHI lagta

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Guna Ko Jod Mein Badalna: Exponents Ka Jaadu

✏️ Solved Examples

🏠 Real Life Mein Exponent Niyam

🧭 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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