Zero Aur Negative Exponent | Class 8 Maths Book 1 Ch 2

📘 Class 8 · Ganita Prakash Book 1Ch 2 · Power PlayPost 5 / 9⏱ 25–30 min

Zero Aur Negative Exponent ➖

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:

3⁴ = ?
2⁵ × 2³ = ?
1 ÷ (1/4) = ?

Answers dekho

81
2⁸ (powers jodo: 5 + 3)
4 (1 ke andar 4 chauthaai hain)

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

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🎬 Kahani: Mithai Ke Tukde

Nani ne 81 tukdon wali barfi ki tray di: “Har baar jo bacha hai, uska ek-tihaai rakh lo, baaki baant do.”

81 → 27 → 9 → 3 → 1. Golu: “Yeh to $3^4 \to 3^3 \to 3^2 \to 3^1 \to$ … aur aakhri 1 tukda? $3^{?}$” 🤔

Meenu: “Aur 1 tukde ka bhi ek-tihaai karein to? $\frac{1}{3}$ tukda! Uski power kya hogi?”

Golu: “Power to ‘kitni baar guna’ hai. Zero baar guna? Minus baar guna?? Yeh to bekaar baat hai!” Kya sach mein? Post ke end tak Golu khud in tukdon ki power batayega. 🍬

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📖 Concept: Bhaag Karo, Power Ghataao

💡 Is post ki ek badi baat: Power line pe daayein chalna = base se guna, baayein chalna = base se bhaag. Power 0 aur negative powers koi naya jaadu nahi: bas line ko baayein badhaaya, aise ki saare purane niyam sahi rahein.

Step 1: Bhaag ka niyam

$3^5 \div 3^2 = \dfrac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$. Upar-neeche do-do 3 kat gaye, teen 3 bache: $3^3$.

$n^a \div n^b = n^{a-b} \quad (n \neq 0)$

WHY ghataana? Har neeche wala base ek upar wale ko kaat deta hai. a mein se b kat gaye → a − b bache. Guna mein ginti judti thi, bhaag mein ghatti hai.

WHY n ≠ 0? 0 se bhaag kabhi defined nahi hota.

Step 2: Power 0 = 1 kyun? (do raaste, ek jawab)

Raasta 1 (niyam se): $3^4 \div 3^4$. Koi bhi number khud se bhaag = 1. Niyam se: $3^{4-4} = 3^0$. To $3^0 = 1$.

Raasta 2 (pattern se): 81, 27, 9, 3: har kadam ÷ 3 aur power 1 kam. 3 ÷ 3 = 1, power 1 → 0. To $3^0 = 1$.

$n^0 = 1 \quad (n \neq 0)$

WHY “zero baar guna” = 1, 0 nahi? Guna ki shuruaat 1 se hoti hai (1 kuch nahi badalta). “Kuch bhi guna nahi kiya” = shuruaati 1 hi bacha. Jaise jod ki shuruaat 0 se: “kuch nahi joda” = 0.

Step 3: Negative power = utni baar bhaag

Pattern aage: 1 ÷ 3 = $\frac{1}{3}$ → $3^{-1}$. Phir $\frac{1}{9}$ → $3^{-2}$.

$n^{-a} = \dfrac{1}{n^a}, \qquad \dfrac{1}{n^{-a}} = n^a$

Niyam se check: $3^2 \div 3^5 = 3^{2-5} = 3^{-3}$. Lamba: $\frac{3 \times 3}{3 \times 3 \times 3 \times 3 \times 3} = \frac{1}{27}$ ✅ Dono same!

WHY minus = ulta? Positive power: 1 ko utni baar guna. Negative power: 1 ko utni baar bhaag. Number negative nahi hota: $3^{-2} = \frac{1}{9}$ positive hai!

Step 4: Power line pe chalo

5 ki power line: 5^−3 = 1/125 se 5^3 = 125 tak, beech mein 5^0 = 1

Daayein = × 5, baayein = ÷ 5. Beech mein $5^0 = 1$ (shuruaat ka bindu).

$5^3 \div 5^{-1} = 5^{3-(-1)} = 5^4$: line pe $5^{-1}$ se $5^3$ tak 4 kadam = 5 × 5 × 5 × 5 = 625 guna. ✅

WHY line achhi? “Kitne guna bada?” = kitne kadam ka farak. $5^2$ (25) aur $5^{-1}$ (1/5): 3 kadam → 125 guna.

Step 5: Do tareeke: fraction banao ya niyam lagao

Sawaal: $2^{-3} \times 2^{5}$ = ?

Tareeka A (fraction): $\frac{1}{8} \times 32 = 4$.

Tareeka B (niyam): $2^{-3+5} = 2^2 = 4$.

Kab kaunsa? Powers chhoti aur value chahiye → A (samajh pakki hoti hai). Powers badi ya letters (x⁻⁷) → B (tez aur saaf). Shuru mein dono se karke milaao: yahi best check hai!

Saare niyam ab har integer power pe: $(2^{-1})^3 = 2^{-3}$, $10^{-2} \div 10^{-5} = 10^{3}$.

🪜 Atak Gaye? Aise Socho

Negative power dekh ke daro mat: bolo “1 by …”. 2⁻³ = “1 by 2³” = 1/8.
Power 0? Jawab 1 (base 0 na ho). Bas!
Bhaag ke sawaal mein ghataav dhyaan se: 3 − (−1) = 3 + 1 = 4.
Atak gaye to power line banao aur kadam gino.
Jawab ko fraction se dobara nikaal ke check karo.

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🔐 Deep Concept: Maths Ke Niyam “Banaaye” Kaise Jaate Hain?

Golu ki aapatti sahi thi: “$3^0$ = 3 ko zero baar guna” ka seedha matlab nahi banta. To ganitagyon ne kya kiya? Unhone poocha: “Iska kya matlab rakhein ki purane niyam toote nahi?”

Niyam tha $n^a \div n^b = n^{a-b}$. Agar a = b ho, to baayein taraf 1 hai, daayein $n^0$. Niyam bachaana hai to $n^0$ = 1 hi rakhna padega. Koi aur choice nahi!

Isi tarah $n^{-a}$: $n^0 \div n^a = n^{-a}$, aur baayein $1 \div n^a$. To $n^{-a} = \frac{1}{n^a}$ hi hona chahiye.

Is soch ka naam: “pattern ko aage badhao” (extension). Negative numbers bhi aise hi aaye the: 3 − 5 ka jawab chahiye tha, to number line ko 0 se baayein badhaaya.

Jahan niyam ruk jaata hai: n = 0. $0^a \div 0^a = 0 \div 0$, jo defined nahi. Isliye $0^0$ aur $0^{-2}$ ko Class 8 mein chhod dete hain. ⚠️

Unifying idea: maths mein nayi cheez tab “sahi” maani jaati hai jab woh purane sach ke saath fit ho. Power line ka baayein hissa isi tarah bana. 🎯

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✏️ Solved Examples

Example 1 (Easy): Value: 4⁻²

“1 by 4²” = $\frac{1}{16}$.

Check: 4 ki line: 16, 4, 1, 1/4, 1/16. 4² se 4 kadam baayein → 4⁻² = 1/16 ✅

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Example 2 (Medium): Ek power mein: 7³ × 7⁻⁵ × 7²

Powers jodo: 3 − 5 + 2 = 0 → 7⁰ = 1.

Check: 343 × (1/16,807) × 49 = 16,807/16,807 = 1 ✅ (jitne guna, utne hi bhaag)

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Example 3 (NCERT style): Simplify: (2⁻³ × 2⁵) ÷ 2⁻¹

$2^{-3+5} = 2^2$. Phir $2^2 \div 2^{-1} = 2^{2-(-1)} = 2^3 = $ 8.

Check (Tareeka A): (1/8 × 32) ÷ (1/2) = 4 ÷ 1/2 = 4 × 2 = 8 ✅

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Golu ka jawab: “Aakhri poora tukda 3⁰ = 1, aur uska ek-tihaai 3⁻¹ = 1/3. Nani, ab mujhe 3² = 9 tukde do!” 😋

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🏠 Real Life Mein Negative Powers

📏 Chhote naap: 1 cm = 10⁻² m, 1 mm = 10⁻³ m. Scale pe har chhota nishaan!
💊 Dawai: 500 mg = 500 × 10⁻³ g = 0.5 g.
🦠 Bacteria: lambai ≈ 10⁻⁶ m (micrometre). Microscope ke bina nahi dikhta.
📸 Camera shutter: 1/1000 second = 10⁻³ s, isliye udti chidiya bhi saaf photo mein.
🍕 Baar-baar aadha: 1 pizza → 2⁻¹, 2⁻², 2⁻³: aadha, chauthaai, aathvaan hissa.

Negative power = “bahut chhota”, negative number nahi. Yaad rakho!

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Same base ka bhaag	✅ Powers ghataao	n^a−b
Non-zero base, power 0	✅ Jawab 1	n⁰ = 1
Negative power, value chahiye	✅ “1 by” banao	n^−a = 1/n^a
Letters/badi powers	✅ Seedha niyam (Tareeka B)	Fraction lambe ho jaate
Base 0, power 0 ya negative	❌ Mat lagao	0 se bhaag

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❌→✅ Worked Mistake

Sawaal: 2⁻³ = ?

❌ Galat: “2⁻³ = −8” ya “= −6”.

Galti kya hai? Minus ko number ka sign samajh liya (−8), ya exponent ko guna kar diya (2 × −3). Negative power ka matlab: utni baar bhaag.

✅ Sahi: 2⁻³ = 1 ÷ 2³ = 1/8. Check: line pe 8, 4, 2, 1, 1/2, 1/4, 1/8 → 2³ se 6 kadam baayein ✅

Doosri common galti: 5⁰ = 0 likhna. Nahi, 1! (0⁵ = 0 hota hai, ulta mat karo.)

🧠 Rule of thumb: Negative power = “1 by”. Positive base ki har power positive hoti hai.

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📝 10 MCQs

Q1. 9⁰ = ?

(a) 0

(b) 9

(c) 1

(d) Undefined

Solution dekho

Sahi jawab: (c)

(a) galat: Power 0 ka jawab 0 nahi; woh 0⁹ jaisa hai.
(b) galat: 9 tab aata jab power 1 hoti.
(c) sahi: Non-zero base, power 0 = 1 ✅
(d) galat: Sirf base 0 ho tab dikkat hoti.

Q2. 5⁻² = ?

(a) −25

(b) 1/25

(c) −10

(d) 25

Solution dekho

Sahi jawab: (b)

(a) galat: Minus ko number ka sign samajh liya.
(b) sahi: Negative power = “1 by 5²” = 1/25 ✅
(c) galat: 5 × (−2): exponent ko guna kar diya.
(d) galat: Minus hi bhool gaye, ulta nahi kiya.

Q3. 6⁷ ÷ 6⁴ = ?

(a) 6³

(b) 6¹¹

(c) 1³

(d) 6²⁸

Solution dekho

Sahi jawab: (a)

(a) sahi: Chaar 6 kate, teen bache → 6³ ✅
(b) galat: Bhaag mein jod diya; guna wala niyam lagaaya.
(c) galat: Base bhi bhaag kar diya; base wahi rehta.
(d) galat: Powers guna kar diye, koi tark nahi.

Q4. 10⁻³ = ?

(a) 0.001

(b) −1,000

(c) 0.0001

(d) −30

Solution dekho

Sahi jawab: (a)

(a) sahi: 1 ÷ 10³ = 1/1000 = 0.001 ✅
(b) galat: Minus ko number ka sign samajh liya.
(c) galat: Ek zero zyada laga diya (10⁻⁴).
(d) galat: 10 × (−3): exponent ko guna kar diya.

Q5. 2³ × 2⁻³ = ?

(a) 0

(b) 1

(c) 2⁻⁹

(d) 64

Solution dekho

Sahi jawab: (b)

(a) galat: Power 0 hui, par 2⁰ = 1, 0 nahi.
(b) sahi: 3 + (−3) = 0 → 2⁰ = 1 ✅
(c) galat: Powers guna kar diye; guna mein jodte hain.
(d) galat: Minus bhool ke 2⁶ = 64.

Q6. 1/7⁻² = ?

(a) 49

(b) 1/49

(c) −49

(d) −14

Solution dekho

Sahi jawab: (a)

(a) sahi: Ulte ka ulta = 7² = 49 ✅
(b) galat: Sirf ek baar ulta kiya.
(c) galat: Sign lagaa diya, jo hota hi nahi.
(d) galat: 7 × (−2): exponent ko guna maan liya.

Q7. (3⁻²)² = ?

(a) 3⁰

(b) 3⁻⁴

(c) 3⁴

(d) 1/9

Solution dekho

Sahi jawab: (b)

(a) galat: −2 + 2 jod diya; power ki power mein guna.
(b) sahi: Block 2 baar: −2 × 2 = −4 ✅
(c) galat: Minus sign gayab kar diya.
(d) galat: Yeh 3⁻² hai, ek baar square bhool gaye.

Q8. Power line pe 4⁻¹ se 4² tak kitne guna?

(a) 3 guna

(b) 16 guna

(c) 64 guna

(d) 4 guna

Solution dekho

Sahi jawab: (c)

(a) galat: Kadam gine (3), par har kadam × 4 hai.
(b) galat: Sirf 4² ki value dekhi.
(c) sahi: 3 kadam → 4³ = 64 guna ✅
(d) galat: Sirf ek kadam gina, teen hain.

Q9. Kaun sa sabse bada?

(a) 2⁻¹

(b) 2⁰

(c) 2⁻³

(d) (1/2)²

Solution dekho

Sahi jawab: (b)

(a) galat: 2⁻¹ = 1/2, jo 1 se chhota.
(b) sahi: 2⁰ = 1, baaki sab fraction ✅
(c) galat: 2⁻³ = 1/8, sabse chhota.
(d) galat: (1/2)² = 1/4, jo 1 se chhota.

Q10. x⁵ ÷ x⁸ = ?

(a) x³

(b) x⁻³

(c) −x³

(d) x¹³

Solution dekho

Sahi jawab: (b)

(a) galat: Ulta ghataaya (8 − 5).
(b) sahi: 5 − 8 = −3 → x⁻³ = 1/x³ ✅
(c) galat: Minus ko power se bahar le aaye.
(d) galat: Bhaag mein powers jod diye, ghataana tha.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. 2⁰ = 1 ratna padega?

Nahi, samjho: 8, 4, 2, … har baar ÷ 2 → agla 1. Wahi 2⁰. Ya 2³ ÷ 2³ = 1.

2. Negative power hamesha chhota number deti hai?

Base 1 se bada ho to haan. Par (1/2)⁻³ = 2³ = 8, bada! Ulta ka ulta.

3. (−2)⁻² kya hai?

1/(−2)² = 1/4. Even power ne minus khatam kar diya.

4. Kya niyam fractions pe chalte hain?

Haan: (2/3)⁻¹ = 3/2. (a/b)⁻ⁿ = (b/a)ⁿ.

5. 1⁻¹⁰⁰?

1/1¹⁰⁰ = 1. 1 ki har power 1.

6. Yeh scientific notation mein kaam aayega?

Bilkul! 0.00045 = 4.5 × 10⁻⁴. Agla post: Post 6 — Scientific Notation Kya Hai?.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy Value: 2⁻⁴

Solution dekho

1/16. Check: 2⁴ = 16, ulta ✅

Q2. Easy Value: 15⁰

Solution dekho

1. WHY: non-zero base, power 0.

Q3. Easy Ek power mein: 3⁹ ÷ 3⁴

Solution dekho

3⁵. WHY: chaar 3 kat gaye, paanch bache.

Q4. Easy 10⁻² decimal mein?

Solution dekho

0.01. Check: 1/100 = 0.01 ✅

Q5. Medium Simplify: 5⁻² × 5⁶

Solution dekho

5⁴ = 625. Check: (1/25) × 15,625 = 625 ✅

Q6. Medium Positive power mein: 1/a⁻⁷

Solution dekho

a⁷. WHY: ulte ka ulta.

Q7. Medium Value: (1/3)⁻²

Solution dekho

(3/1)² = 9. Check: (1/3)² = 1/9, aur 1/(1/9) = 9 ✅

Q8. Medium Ek power mein: 10⁻⁴ ÷ 10⁻⁶

Solution dekho

−4 − (−6) = 2 → 10². Check: 0.0001 ÷ 0.000001 = 100 ✅

Q9. NCERT style Equivalent form: (i) 3⁻³ (ii) (−4)⁻³ (iii) 10⁻⁸

Solution dekho

(i) 1/27 (ii) 1/(−64) = −1/64 (iii) 1/10⁸. Check (ii): odd power, to minus bacha ✅

Q10. NCERT style Simplify, power mein: 6⁻³ × 6⁸ × 6⁻²

Solution dekho

−3 + 8 − 2 = 3 → 6³ = 216. Check: line pe 3 kadam baayein, 8 daayein, 2 baayein = 3 daayein ✅

Q11. NCERT style Simplify: m⁴ × m⁻⁹

Solution dekho

m⁻⁵ = 1/m⁵. Check: m = 2 → 16 × (1/512) = 1/32 = 1/2⁵ ✅

Q12. NCERT style 2 ki power line mein 2⁶, 2⁻² se kitne guna bada?

Solution dekho

6 − (−2) = 8 kadam → 2⁸ = 256 guna. Check: 64 ÷ (1/4) = 256 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. 2^x = 1/32. x?

Solution dekho

32 = 2⁵, aur 1/2⁵ = 2⁻⁵ → x = −5.

Q2. (2⁻¹ + 3⁻¹)⁻¹ = ?

Solution dekho

1/2 + 1/3 = 5/6. Ulta: 6/5. Dhyaan: (a + b)⁻¹ ≠ a⁻¹ + b⁻¹ (jaise 5/6 ka ulta 5 nahi).

Q3. Riya: “3⁻² < 0, kyunki power negative hai.” Galti samjhao.

Solution dekho

3⁻² = 1/9 > 0. Negative power “bhaag” karti hai, sign nahi badalti. Positive number ko kitna bhi bhaag do, positive hi rahega.

Q4. 1 m ke dhaage ko baar-baar aadha karte hain. Kitni baar ke baad lambai 1 mm se kam hogi?

Solution dekho

2⁻ⁿ < 10⁻³ → 2ⁿ > 1,000 → n = 10 (1/1,024 m ≈ 0.98 mm). Check: n = 9 pe 1/512 m ≈ 1.95 mm, abhi zyada ✅

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🏆 Olympiad Corner

Q1. (5⁻¹ × 5³)² ÷ 5⁴ = ?

Solution dekho

(5²)² = 5⁴; ÷ 5⁴ = 1.

Q2. 4⁻³ × 8² = 2^?

Solution dekho

Base 2 banao: 2⁻⁶ × 2⁶ = 2⁰ → 0.

Q3. Agar 3ⁿ = 1/81 to 3ⁿ⁺²?

Solution dekho

n = −4, to 3⁻² = 1/9. Ya seedha: 1/81 × 9 = 1/9.

Q4. Kaun bada: 2⁻¹⁰ ya 10⁻³?

Solution dekho

2¹⁰ = 1,024 > 1,000, to 1/1,024 < 1/1,000. 10⁻³ bada. (Bade number ka ulta chhota.)

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💗 CharuMam Motivation

Golu ne kaha tha “zero baar guna? bekaar baat!” Par jab usne pattern pe bharosa kiya, to ek naya sach mil gaya. Bade ganitagya bhi aise hi sochte the: “agar pattern aage chale to?” 🔍

Aapke “yeh to ajeeb hai!” wale sawaal hi aapko aage le jaate hain. Poochhte raho, dhoondhte raho! 🚀

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🟢 Master Recap

💡 Chaabi: Power line pe daayein chalna = base se guna, baayein chalna = base se bhaag. Power 0 aur negative powers koi naya jaadu nahi: bas line ko baayein badhaaya, aise ki saare purane niyam sahi rahein.
n^a ÷ n^b = n^a−b (n ≠ 0): kate hue base ghatao.
n⁰ = 1 (n ≠ 0): guna ki shuruaat 1 se.
n^−a = 1/n^a; 1/n^−a = n^a.
Negative power ≠ negative number.
Power line: daayein × base, baayein ÷ base; kadam = guna ka farak.
Value chahiye → fraction; letters/badi powers → niyam.
Naye niyam aise bane ki purane niyam toote nahi.

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👉 Aage Kya?

10 ki positive aur negative powers se ab hum bahut bade aur bahut chhote number ko chhota likh sakte hain: Suraj ki doori ho ya virus ki lambai! Agle post mein: scientific notation. 🔭

← Pichla: Post 4 — Combinations Aur Powers 📚 Chapter 2 ke saare posts Agla: Post 6 — Scientific Notation Kya Hai? →

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

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Mithai Ke Tukde

📖 Concept: Bhaag Karo, Power Ghataao

Step 1: Bhaag ka niyam

Step 2: Power 0 = 1 kyun? (do raaste, ek jawab)

Step 3: Negative power = utni baar bhaag

Step 4: Power line pe chalo

Step 5: Do tareeke: fraction banao ya niyam lagao

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Maths Ke Niyam “Banaaye” Kaise Jaate Hain?

✏️ Solved Examples

🏠 Real Life Mein Negative Powers

🧭 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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