NCERT Exemplar ke “Exponents and Powers” unit ke topics pe poori practice, ek hi post mein: base aur exponent, exponents ke niyam, zero aur negative power, aur bahut bade-chhote numbers ka standard form. Pehle chhota recap, phir 10 MCQs, fill in the blanks, true/false, short aur long answer, HOTS, aur aakhir mein answer key. Har sawaal ka hal Hinglish mein step by step hai. Yeh post aapke current chapter Book 1 Ch 2: Power Play ke saath padhiye: chapter se samjho, yahan se practice karo. ⚡
🧠 Concept Recap: Pehle Yeh Yaad Kar Lo
📖 Pehle Book 1 Ch 2 padho: Power Play (hub)
📏 Pehle yeh shabd samjho:
- Base aur exponent = 2⁵ mein neeche wala bada number 2 base hai (jise baar-baar guna karna hai), aur upar wala chhota number 5 exponent (power, ghaat) hai: kitni baar guna karna hai. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Ise padhte hain “2 ki power 5”.
- Integer = poore number, zero aur minus wale bhi: …, −3, −2, −1, 0, 1, 2, 3, …
- Negative = zero se kam number, jaise −3 (thermometer mein zero se neeche)
- Fraction (bhinn) = 1/8 matlab 1 ko 8 barabar hisson mein baanta, unme se ek hissa. Upar wala numerator, neeche wala denominator.
- Reciprocal (ulta) = fraction ko ulat do: 2/3 ka reciprocal 3/2; 5 (yaani 5/1) ka reciprocal 1/5. Number × uska reciprocal = 1.
- Decimal = point wala number: 0.01 = 1/100, 0.001 = 1/1000
- ≠ = “barabar nahi”. a ≠ 0 matlab a zero nahi hai.
- a, m, n jaise letters = kisi bhi number ki jagah rakhe gaye “khaali dabbe”. Niyam letters mein isliye likhte hain ki woh har number pe chale.
- > aur < = > = bada hai (5 > 3), < = chhota hai (3 < 5); ≤ = chhota ya barabar
- Standard form = bahut bade ya bahut chhote number ko chhota likhne ka tareeka, jaise 384000 = 3.84 × 10⁵ (neeche point 4 dekho)
- Units digit = number ki aakhri (ikai wali) digit: 2187 ki units digit 7
- Simplify = lambe hisaab ko chhota karke ek jawab tak laana
Worked example (3⁴ ko kholna): 3⁴ = 3 × 3 × 3 × 3. Ek-ek karke: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81. To 3⁴ = 81.
1. Power ka matlab
Ek hi number ko baar-baar guna karna ho to chhota likhte hain: 5 × 5 × 5 = 5³ (“5 ki power 3” ya “5 ka cube”). 10 × 10 × 10 × 10 = 10⁴ = 10000 (1 ke aage 4 zero).
- Negative base: (−2)² = (−2) × (−2) = 4 (minus × minus = plus), aur (−2)³ = 4 × (−2) = −8. Yaani even power (2, 4, 6…) pe jawab plus, odd power (1, 3, 5…) pe minus.
- Bracket ka dhyaan: (−3)² = 9, par −3² = −(3 × 3) = −9. Bracket na ho to power sirf 3 pe lagti hai, minus baad mein.
- (−1) ki power: (−1) ki even power = 1, odd power = −1.
2. Exponents ke niyam (a, b ≠ 0; m, n koi bhi integer)
| Niyam | Matlab (shabdon mein) | Example |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Same base ka guna: powers jodo | 2³ × 2⁴ = 2⁷ = 128 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Same base ka bhaag: powers ghatao | 5⁶ ÷ 5⁴ = 5² = 25 |
| (aᵐ)ⁿ = aᵐⁿ | Power ki power: powers ka guna | (3²)³ = 3⁶ = 729 |
| aᵐ × bᵐ = (a × b)ᵐ | Same power, alag base: base ka guna | 2³ × 5³ = 10³ = 1000 |
| aᵐ ÷ bᵐ = (a ÷ b)ᵐ | Same power, alag base: base ka bhaag | 12² ÷ 4² = 3² = 9 |
| a⁰ = 1 | Zero power hamesha 1 (a ≠ 0) | 7⁰ = 1, (−9)⁰ = 1 |
| a⁻ⁿ = 1/aⁿ | Minus power = reciprocal (ulta) ki plus power | 2⁻³ = 1/2³ = 1/8 |
a⁰ = 1 kyun? 5³ ÷ 5³ = 125 ÷ 125 = 1. Niyam se 5³ ÷ 5³ = 5³⁻³ = 5⁰. Dono ek hi cheez hain, isliye 5⁰ = 1.
Minus power kyun ulta? Pattern dekho: 10³ = 1000, 10² = 100, 10¹ = 10, 10⁰ = 1. Har kadam pe 10 se bhaag. Aage bhi waise hi: 10⁻¹ = 1/10, 10⁻² = 1/100, 10⁻³ = 1/1000 = 0.001.
- Fraction ki minus power: fraction ko ulta karo aur power plus kar do, phir upar aur neeche dono ka square (ya jo power ho) karo: (2/3)⁻² = (3/2)² = (3 × 3)/(2 × 2) = 9/4.
- Dhyaan: niyam sirf same base (guna/bhaag) ya same power pe chalte hain. 2³ + 2⁴ mein powers nahi jodte! (2³ + 2⁴ = 8 + 16 = 24, jabki 2⁷ = 128.)
3. Standard form (scientific notation)
Bahut bada ya bahut chhota number k × 10ⁿ ki shakal mein, jahan 1 ≤ k < 10 (k 1 ya usse bada, par 10 se chhota) aur n ek integer.
- Bada number: point ko left le jao jab tak aage ek hi digit bache; jitni jagah khiska, utni plus power. 384000 → 3.84 (point 5 jagah left) → 3.84 × 10⁵.
- Chhota number: point ko right le jao; jitni jagah khiska, utni minus power. 0.0007 → 7 (point 4 jagah right) → 7 × 10⁻⁴.
- Expanded form: 3405.2 = 3 × 10³ + 4 × 10² + 0 × 10¹ + 5 × 10⁰ + 2 × 10⁻¹ (har digit uski jagah ki power of 10 se).
- Tulna (compare): pehle 10 ki power dekho: jiski power badi, woh bada. Power barabar ho, tab k dekho. 2 × 10⁸ > 9 × 10⁷.
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📝 Section A: 10 MCQs
Har sawaal ke 4 options mein se ek sahi chuno. Pehle khud socho, phir “Hal dekho” kholo.
Q1. (−3)⁴ ki value kya hai?
Hal (step by step) dekho
Sahi jawab: (b) 81
Step 1: (−3)⁴ = (−3) × (−3) × (−3) × (−3). Yeh 4 ke saath guna nahi hai (−12 galat hai).
Step 2: ek-ek karke: (−3) × (−3) = 9 (minus × minus = plus), 9 × (−3) = −27, (−27) × (−3) = 81.
Shortcut: power 4 even hai, to jawab plus: 81.
Q2. 5⁻² = ?
Hal (step by step) dekho
Sahi jawab: (c) 1/25
Niyam: a⁻ⁿ = 1/aⁿ. To 5⁻² = 1/5² = 1/(5 × 5) = 1/25.
Dhyaan: minus power ka matlab minus jawab nahi, balki ulta (reciprocal). Aur 5 × (−2) = −10 wala guna bilkul galat hai.
Q3. 2⁵ × 2³ ÷ 2⁴ = ?
Hal (step by step) dekho
Sahi jawab: (a) 2⁴
Step 1: base sab jagah 2 hai, to niyam lagenge.
Step 2: guna pe powers jodo: 2⁵ × 2³ = 2⁵⁺³ = 2⁸.
Step 3: bhaag pe powers ghatao: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16.
Check: 32 × 8 = 256, 256 ÷ 16 = 16 ✅
Q4. 3⁰ + 4⁰ + 5⁰ = ?
Hal (step by step) dekho
Sahi jawab: (c) 3
Har non-zero number ki power 0 = 1. To 3⁰ = 1, 4⁰ = 1, 5⁰ = 1.
Jod: 1 + 1 + 1 = 3. (0 nahi! Power 0 ka matlab jawab 0 nahi hota.)
Q5. 0.000052 ka standard form kaunsa hai?
Hal (step by step) dekho
Sahi jawab: (b) 5.2 × 10⁻⁵
Step 1: point ko right khiskao jab tak 1 aur 10 ke beech ka number (5.2) na bane: 0.000052 → 5.2. Point ne 4 zero aur 5 paar kiye = 5 jagah right gaya.
Step 2: right khiska to minus power: 5.2 × 10⁻⁵.
Dhyaan: 52 × 10⁻⁶ aur 0.52 × 10⁻⁴ ki value bhi same hai, par woh standard form nahi, kyunki 52 aur 0.52, 1 aur 10 ke beech nahi. Isliye sahi jawab sirf 5.2 × 10⁻⁵.
Q6. (2/5)⁻³ = ?
Hal (step by step) dekho
Sahi jawab: (b) 125/8
Step 1: minus power hatane ke liye fraction ulta karo: (2/5)⁻³ = (5/2)³.
Step 2: upar aur neeche dono ka cube: 5³ = 125 aur 2³ = 8.
Jawab: 125/8. (8/125 tab hota jab power +3 hoti.)
Q7. Agar 3ˣ⁺¹ = 81 ho, to x = ?
Hal (step by step) dekho
Sahi jawab: (b) 3
Step 1: 81 ko 3 ki power mein likho: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81, yaani 81 = 3⁴.
Step 2: ab 3ˣ⁺¹ = 3⁴. Base same, to powers barabar: x + 1 = 4.
Step 3: x = 4 − 1 = 3. Check: 3³⁺¹ = 3⁴ = 81 ✅
Q8. (−1)¹⁰¹ + (−1)¹⁰⁰ = ?
Hal (step by step) dekho
Sahi jawab: (b) 0
(−1) ki odd power = −1, even power = +1.
101 odd hai → (−1)¹⁰¹ = −1. 100 even hai → (−1)¹⁰⁰ = 1.
Jod: −1 + 1 = 0.
Q9. In mein se sabse bada kaun hai?
Hal (step by step) dekho
Sahi jawab: (c) 3⁴
Har ek kholo: 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64; 6² = 6 × 6 = 36; 3⁴ = 3 × 3 × 3 × 3 = 81; 4³ = 4 × 4 × 4 = 64.
Sabse bada 3⁴ = 81. Seekh: base aur power ki jagah badalne pe (2⁶ aur 6²) jawab badal jaata hai.
Q10. Suraj aur Dharti ki doori lagbhag 1.5 × 10⁸ km hai. Ise aam (usual) form mein likho.
Hal (step by step) dekho
Sahi jawab: (b) 150000000 km
Step 1: 10⁸ = 1 ke aage 8 zero = 100000000.
Step 2: 1.5 × 100000000: point ko 8 jagah right khiskao. 1.5 mein point ke baad ek digit (5) hai, to 5 ke baad 7 zero aur lagenge: 150000000.
Jawab: 150000000 km (15 crore km). Check: 150000000 ÷ 100000000 = 1.5 ✅
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✍️ Section B: Fill in the Blanks
Khaali jagah (_____) mein sahi shabd ya number bharo.
Q1. 10⁰ = _____
Jawab aur kaaran
Jawab: 1
Koi bhi non-zero number ki power 0 = 1. 10⁰ = 1 (pattern: 10² = 100, 10¹ = 10, 10⁰ = 1, har baar 10 se bhaag).
Q2. 7⁻¹ = _____
Jawab aur kaaran
Jawab: 1/7
a⁻¹ = 1/a, yaani number ka reciprocal. 7⁻¹ = 1/7.
Q3. (2³)² = 2 ki power _____
Jawab aur kaaran
Jawab: 6
Power ki power mein powers ka guna: (2³)² = 2 ki power “3 guna 2” yaani 2⁶ = 64. Check: 2³ = 8, aur 8² = 8 × 8 = 64 ✅
Q4. 4⁻³ × 4⁵ = _____
Jawab aur kaaran
Jawab: 16
Same base, guna, to powers jodo: −3 + 5 = 2. To 4⁻³ × 4⁵ = 4² = 4 × 4 = 16.
Q5. 0.000000001 m = 1 × 10 ki power _____ m
Jawab aur kaaran
Jawab: −9
0.000000001 mein point ke baad 9 jagah hain (8 zero aur phir 1). Point ko 9 jagah right khiskaaya to 1 bana, isliye 1 × 10⁻⁹ m. (Isi ko 1 nanometre kehte hain.)
Q6. Standard form k × 10ⁿ mein k hamesha _____ hota hai.
Jawab aur kaaran
Jawab: 1 ya usse bada, par 10 se chhota (1 ≤ k < 10)
k mein point se pehle sirf ek non-zero digit hoti hai: 3.84, 7, 1.5 sahi; 38.4 ya 0.384 nahi.
Q7. (−2)⁻³ = _____
Jawab aur kaaran
Jawab: −1/8
Step 1: (−2)⁻³ = 1/(−2)³.
Step 2: (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8 (power 3 odd, to minus).
Step 3: 1/(−8) = −1/8.
Q8. 6⁵ ÷ 3⁵ = _____
Jawab aur kaaran
Jawab: 32
Power same (5) hai, to base ka bhaag: (6 ÷ 3)⁵ = 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
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✅❌ Section C: True / False
Batao statement sahi (True) hai ya galat (False). Galat ho to use sahi karke likho.
Q1. 2³ = 3²
Jawab aur kaaran
❌ False
Galat. 2³ = 2 × 2 × 2 = 8, aur 3² = 3 × 3 = 9. 8 aur 9 barabar nahi. Base aur power ki jagah badal do to jawab badal jaata hai.
Q2. 2³ × 3² = 6⁵
Jawab aur kaaran
❌ False
Galat. Baayan (= ke left wala hissa): 8 × 9 = 72. Daayan (right wala hissa): 6⁵ = 6 × 6 × 6 × 6 × 6 = 7776. Niyam aᵐ × aⁿ tabhi lagta hai jab base same ho, aur aᵐ × bᵐ tabhi jab power same ho. Yahan dono alag hain, to koi niyam nahi lagta; seedha 72.
Q3. Har non-zero number ki power 0, 1 ke barabar hoti hai.
Jawab aur kaaran
✅ True
Sahi. 9⁰ = 1, (−4)⁰ = 1, (1/2)⁰ = 1. Kaaran: aⁿ ÷ aⁿ = 1, aur niyam se aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰.
Q4. (−5)² aur −5² ki value same hai.
Jawab aur kaaran
❌ False
Galat. (−5)² = (−5) × (−5) = 25. Par −5² mein bracket nahi, to power sirf 5 pe: −(5 × 5) = −25. Sahi: “(−5)² = 25 aur −5² = −25.”
Q5. 10⁻² > 10⁻³
Jawab aur kaaran
✅ True
Sahi. 10⁻² = 1/100 = 0.01 aur 10⁻³ = 1/1000 = 0.001. 0.01, 0.001 ka 10 guna hai, to bada hai. (Minus power mein jitna bada number, utna chhota jawab.)
Q6. 3.4 × 10⁻⁵, 3.4 × 10⁻⁴ se bada hai.
Jawab aur kaaran
❌ False
Galat. 3.4 × 10⁻⁵ = 0.000034 aur 3.4 × 10⁻⁴ = 0.00034. Doosra 10 guna bada hai. Sahi: “3.4 × 10⁻⁴, 3.4 × 10⁻⁵ se bada hai.”
Q7. (2⁻¹ + 3⁻¹)⁻¹ = 6/5
Jawab aur kaaran
✅ True
Sahi. Step 1: 2⁻¹ = 1/2, 3⁻¹ = 1/3. Step 2: 1/2 + 1/3 jodne ke liye dono ka neeche wala number same (6) banao. 1/2 ke upar-neeche dono ko 3 se guna: 1/2 = (1 × 3)/(2 × 3) = 3/6. 1/3 ke upar-neeche dono ko 2 se guna: 1/3 = (1 × 2)/(3 × 2) = 2/6. (Upar-neeche dono ko same number se guna karne pe value nahi badalti.) Ab neeche same hai, to upar wale jodo: 3/6 + 2/6 = 5/6. Step 3: (5/6)⁻¹ = ulta = 6/5 ✅
Q8. 25 × 10⁴ standard form mein likha hai.
Jawab aur kaaran
❌ False
Galat. Standard form mein k, 1 aur 10 ke beech hona chahiye; 25 nahi hai. 25 × 10⁴ = 250000 = 2.5 × 10⁵. Sahi: “250000 ka standard form 2.5 × 10⁵ hai.”
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🟢 Section D: Short Answer Questions
2 se 3 marks wale sawaal. Har step likhne ki aadat daalo.
Q1. Simplify karo: (2⁵ × 2⁻² × 2⁴) ÷ 2³
Step-by-step hal
Step 1: saare base 2 hain, to sirf powers ka hisaab karo.
Step 2: guna wale hisse mein powers jodo: 5 + (−2) + 4 = 7. To 2⁵ × 2⁻² × 2⁴ = 2⁷.
Step 3: ab bhaag: 2⁷ ÷ 2³ = 2⁷⁻³ = 2⁴.
Step 4: 2⁴ = 2 × 2 × 2 × 2 = 16.
Check: 2⁵ = 32, 2⁻² = 1/4, 2⁴ = 16, 2³ = 8. 32 × (1/4) matlab 32/1 × 1/4 = (32 × 1)/(1 × 4) = 32/4 = 8 (upar × upar, neeche × neeche; yaani 32 ka chautha hissa). Phir 8 × 16 = 128, aur 128 ÷ 8 = 16 ✅
Q2. x nikaalo: 5²ˣ⁺¹ ÷ 25 = 125
Step-by-step hal
Step 1: sab kuch 5 ki power mein likho: 25 = 5 × 5 = 5², aur 125 = 5 × 5 × 5 = 5³.
Step 2: sawaal ban gaya: 5²ˣ⁺¹ ÷ 5² = 5³.
Step 3: bhaag mein powers ghatao: 5²ˣ⁺¹⁻² = 5²ˣ⁻¹. To 5²ˣ⁻¹ = 5³.
Step 4: base same (5), to powers barabar: 2x − 1 = 3. (2x matlab 2 guna x, yaani 2 × x.)
Step 5: dono taraf 1 jodo, taaki −1 hat jaaye (taraazu ke dono palde mein same cheez daalo to barabari bani rehti hai): 2x − 1 + 1 = 3 + 1, yaani 2x = 4.
Step 6: 2 guna x = 4, to x akela paane ke liye dono taraf 2 se bhaag do: x = 4 ÷ 2 = 2.
Check: x = 2 pe 2x + 1 = 5, aur 5⁵ = 3125. 3125 ÷ 25 = 125 ✅
Q3. Standard form mein likho: (a) 3,05,00,000 (b) 0.00000408
Step-by-step hal
(a) 30500000: point abhi aakhri mein hai. Use left khiskao jab tak aage sirf 3 bache: 3.0500000 → point 7 jagah left gaya. Aakhri ke zero hata do: 3.05 × 10⁷. Check: 3.05 × 10000000 = 30500000 ✅
(b) 0.00000408: point ko right khiskao jab tak 4.08 na bane: point 6 jagah right gaya (5 zero paar karke 4 ke baad). Right khiska to minus power: 4.08 × 10⁻⁶.
Q4. Value nikaalo: (1/2)⁻² + (1/3)⁻² + (1/4)⁻²
Step-by-step hal
Step 1: har fraction ko ulta karo aur power plus karo: (1/2)⁻² = 2², (1/3)⁻² = 3², (1/4)⁻² = 4².
Step 2: 2² = 4, 3² = 9, 4² = 16.
Step 3: jod: 4 + 9 + 16 = 29.
Q5. Kaun bada hai: 2.7 × 10¹² ya 1.5 × 10¹³? Kaaran do.
Step-by-step hal
Step 1: pehle 10 ki power dekho: 10¹³ ki power (13) badi hai, 10¹² ki (12) chhoti.
Step 2: pakka karne ke liye dono ko same power mein laao: 1.5 × 10¹³ = 1.5 × 10 × 10¹² = 15 × 10¹².
Step 3: ab 15 × 10¹² aur 2.7 × 10¹²: 15 > 2.7.
Jawab: 1.5 × 10¹³ bada hai (lagbhag 5.6 guna, kyunki 15 ÷ 2.7 lagbhag 5.6; matlab ek number doosre ka 5 se thoda zyada guna hai).
Q6. 47053.06 ko powers of 10 ki madad se expanded form mein likho.
Step-by-step hal
Step 1: har digit ki jagah pehchaano: 4 (das hazaar), 7 (hazaar), 0 (sau), 5 (das), 3 (ikai), 0 (dasvaan hissa), 6 (sauvaan hissa).
Step 2: jagah ko power of 10 mein: das hazaar = 10⁴, hazaar = 10³, sau = 10², das = 10¹, ikai = 10⁰, dasvaan = 10⁻¹, sauvaan = 10⁻².
Jawab: 4 × 10⁴ + 7 × 10³ + 0 × 10² + 5 × 10¹ + 3 × 10⁰ + 0 × 10⁻¹ + 6 × 10⁻².
Check: 40000 + 7000 + 0 + 50 + 3 + 0 + 0.06 = 47053.06 ✅
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📘 Section E: Long Answer Questions
4 se 5 marks wale sawaal. Hal mein har step ka kaaran bhi likha hai.
Q1. Simplify karo: (a) (2⁻¹ × 4⁻¹) ÷ 2⁻² (b) {(3/4)⁻¹ − (1/4)⁻¹}⁻¹
Step-by-step hal
(a) Tareeka 1 (fraction se): 2⁻¹ = 1/2, 4⁻¹ = 1/4, 2⁻² = 1/2² = 1/4. Sawaal ban gaya: (1/2 × 1/4) ÷ 1/4.
Fraction ka guna: upar × upar, neeche × neeche. Roti socho: aadhi roti (1/2) ka chautha hissa (1/4) lo, to poori roti ka 1/8 tukda milta hai. Isi tarah 1/2 × 1/4 = (1 × 1)/(2 × 4) = 1/8.
Fraction se bhaag = ulta karke guna: 1/8 ÷ 1/4 matlab “1/8 mein 1/4 kitni baar?” Niyam: doosre fraction (1/4) ko ulta karo (4/1) aur guna karo: 1/8 ÷ 1/4 = 1/8 × 4/1 = (1 × 4)/(8 × 1) = 4/8 = 1/2. (4/8 ke upar-neeche dono ko 4 se bhaag diya.)
(a) Tareeka 2 (niyam se): 4 = 2², to 4⁻¹ = (2²)⁻¹ = 2⁻². Ab 2⁻¹ × 2⁻² ÷ 2⁻² = 2⁻¹⁻²⁺² = 2⁻¹ = 1/2 ✅ Dono tareeke se same jawab.
(b) Step 1: andar ke dono ulte karo: (3/4)⁻¹ = 4/3, (1/4)⁻¹ = 4/1 = 4.
Step 2: ghatao: 4/3 − 4 = 4/3 − 12/3 = −8/3 (4 ko 12/3 likha taaki neeche dono mein 3 ho).
Step 3: bahar wali −1 power: (−8/3)⁻¹ = ulta = 3/(−8) = −3/8.
Q2. Ek lab mein bacteria ki ginti har ghante double ho jaati hai. Shuru mein 500 bacteria hain. (a) 6 ghante baad kitne honge? Jawab standard form mein bhi likho. (b) Kitne poore ghante baad ginti pehli baar 1,00,000 se zyada hogi?
Step-by-step hal
Samjho: har ghante 2 se guna. 1 ghante baad 500 × 2, 2 ghante baad 500 × 2 × 2 = 500 × 2², … n ghante baad 500 × 2ⁿ. (Ise exponential growth kehte hain: har baraabar samay mein ginti same number se guna hoti hai, isliye power lagti hai aur ginti bahut tezi se badhti hai. Power Play mein yahi padha.)
(a) Step 1: 6 ghante: 500 × 2⁶. 2⁶ = 64.
Step 2: 500 × 64 = 32000 bacteria.
Step 3: standard form: 32000 → 3.2 (point 4 jagah left) → 3.2 × 10⁴.
(b) Step 1: aage badhte jao: 7 ghante: 32000 × 2 = 64000 (abhi 1,00,000 se kam).
Step 2: 8 ghante: 64000 × 2 = 128000 (1,00,000 se zyada ✅).
Jawab: 8 ghante baad (tab 1,28,000 bacteria).
Q3. Dharti ka dravyamaan (mass, yaani kitna “maal” hai) lagbhag 5.97 × 10²⁴ kg aur Chaand ka 7.35 × 10²² kg hai. (a) Dono ka kul mass standard form mein nikaalo. (b) Dharti, Chaand se lagbhag kitne guna bhaari hai?
Step-by-step hal
(a) Step 1: jodne ke liye dono ki 10 ki power same karo. 7.35 × 10²² ko 10²⁴ mein badlo: 10²⁴, 10²² se do power bada hai, yaani 100 guna (10 × 10) bada. Power ko 100 guna bada kiya, to value same rakhne ke liye number ko 100 se chhota karo (point 2 jagah left): 7.35 ÷ 100 = 0.0735. To 7.35 × 10²² = 0.0735 × 10²⁴.
Step 2: ab jodo: 5.97 × 10²⁴ + 0.0735 × 10²⁴ = (5.97 + 0.0735) × 10²⁴ = 6.0435 × 10²⁴ kg.
Step 3: 6.0435, 1 aur 10 ke beech hai, to yeh standard form hai: 6.0435 × 10²⁴ kg.
(b) Step 1: bhaag: (5.97 × 10²⁴) ÷ (7.35 × 10²²) = (5.97 ÷ 7.35) × 10²⁴⁻²² = (5.97 ÷ 7.35) × 10².
Step 2: 5.97 ÷ 7.35 ka mota andaaza: 5.97 lagbhag 6, aur 7.35 lagbhag 7.5. 6 ÷ 7.5 = 0.8. Thoda aur sahi karein to 5.97 ÷ 7.35 lagbhag 0.81 aata hai (check: 7.35 × 0.81 = 5.9535, jo 5.97 ke bahut paas hai).
Step 3: 0.81 × 100 = lagbhag 81.
Jawab: Dharti lagbhag 81 guna bhaari hai.
Q4. Ek kaagaz ki sheet 0.016 cm moti hai. (a) Aisi 5000 sheets ka dher (stack) kitna ooncha hoga? Standard form ka use karke hal karo. (b) Ek sheet ko beech se 10 baar modo (har baar motaai double hoti hai). Ab moda hua kaagaz kitna mota hoga?
Step-by-step hal
(a) Step 1: 0.016 = 1.6 × 10⁻² (point 2 jagah right) aur 5000 = 5 × 10³.
Step 2: guna: (1.6 × 5) × (10⁻² × 10³) = 8 × 10⁻²⁺³ = 8 × 10¹ = 80.
Jawab: dher 80 cm ooncha. Check: 0.016 × 5000 = 80 ✅
(b) Step 1: ek baar modne pe motaai × 2, do baar × 2 × 2 = × 2², … 10 baar × 2¹⁰.
Step 2: 2¹⁰ = 1024 (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024).
Step 3: 0.016 × 1024 = 16.384 cm.
Jawab: lagbhag 16.4 cm mota, sirf ek kaagaz se! Yahi doubling ki taaqat hai.
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🧩 Section F: HOTS (Soch Wale Sawaal)
HOTS matlab Higher Order Thinking Skills: aise sawaal jinme formula se aage thoda dimaag lagana padta hai.
Q1. 7¹⁰⁰ ki units digit (aakhri digit) kya hogi?
Soch aur hal dekho
Soch: poora 7¹⁰⁰ nikaalna namumkin, par aakhri digit ka pattern dekho.
7¹ = 7, 7² = 49 (aakhri 9), 7³ = 343 (aakhri 3), 7⁴ = 2401 (aakhri 1), 7⁵ = 16807 (aakhri phir 7). Pattern 7, 9, 3, 1 har 4 power pe dohraata hai.
100 ÷ 4 = 25, bilkul poora bant gaya (shesh 0). Matlab 7¹⁰⁰ pattern ke chauthe number pe aata hai, jaise 7⁴.
Jawab: units digit 1.
Q2. Bina calculator: kaun bada hai, 2³⁰⁰ ya 3²⁰⁰?
Soch aur hal dekho
Soch: dono ko same power mein le aao. 300 = 3 × 100 aur 200 = 2 × 100.
2³⁰⁰ = (2³)¹⁰⁰ = 8¹⁰⁰ (power ki power ka niyam ulta lagaya).
3²⁰⁰ = (3²)¹⁰⁰ = 9¹⁰⁰.
Ab dono ki power 100 same hai, aur 9 > 8, to 9¹⁰⁰ > 8¹⁰⁰.
Jawab: 3²⁰⁰ bada hai.
Q3. Agar 2ˣ = 3 ho, to (a) 2ˣ⁺³ aur (b) 4ˣ ki value kya hogi?
Soch aur hal dekho
(a) Niyam ulta lagao: 2ˣ⁺³ = 2ˣ × 2³ = 3 × 8 = 24.
(b) 4 = 2², to 4ˣ = (2²)ˣ = 2²ˣ = (2ˣ)² = 3² = 9.
Seekh: x ki value pata na ho tab bhi niyamon se jawab nikal aata hai!
Q4. Riya ne likha: 2⁵ + 2⁵ = 2¹⁰. Kya yeh sahi hai? Agar nahi, to 2⁵ + 2⁵ ko 2 ki power ke roop mein sahi likho.
Soch aur hal dekho
Galti pakdo: powers jodne ka niyam sirf guna pe hai (2⁵ × 2⁵ = 2¹⁰), jod pe nahi.
Sahi tareeka: 2⁵ + 2⁵ matlab do baar 2⁵, yaani 2 × 2⁵ = 2¹ × 2⁵ = 2⁶.
Check: 32 + 32 = 64, aur 2⁶ = 64 ✅ Jabki 2¹⁰ = 1024, bahut bada.
Jawab: nahi; 2⁵ + 2⁵ = 2⁶.
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🔑 Answer Key (Ek Nazar Mein)
Pehle saare sawaal khud karo, phir yahan milaao. Jahan galti ho, upar us sawaal ka step-by-step hal dobara padho.
| Section | Answers |
|---|---|
| A. MCQ | 1: (b) · 2: (c) · 3: (a) · 4: (c) · 5: (b) · 6: (b) · 7: (b) · 8: (b) · 9: (c) · 10: (b) |
| B. Fill in the Blanks | 1: 1 · 2: 1/7 · 3: 6 · 4: 16 · 5: −9 · 6: 1 ya usse bada, par 10 se chhota (1 ≤ k < 10) · 7: −1/8 · 8: 32 |
| C. True / False | 1: F · 2: F · 3: T · 4: F · 5: T · 6: F · 7: T · 8: F |
| D. Short Answer | 1: 16 · 2: x = 2 · 3: (a) 3.05 × 10⁷ (b) 4.08 × 10⁻⁶ · 4: 29 · 5: 1.5 × 10¹³ · 6: 4 × 10⁴ + 7 × 10³ + 0 × 10² + 5 × 10¹ + 3 × 10⁰ + 0 × 10⁻¹ + 6 × 10⁻² |
| E. Long Answer | 1: (a) 1/2 (b) −3/8 · 2: (a) 32000 = 3.2 × 10⁴ (b) 8 ghante · 3: (a) 6.0435 × 10²⁴ kg (b) lagbhag 81 guna · 4: (a) 80 cm (b) 16.384 cm |
| F. HOTS | 1: 1 · 2: 3²⁰⁰ · 3: (a) 24 (b) 9 · 4: Nahi; 2⁶ |
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👩🏫 CharuMam Ka Tip
Exponents ke sawaal mein pehle sab kuch same base mein badlo (4 = 2², 8 = 2³, 25 = 5², 81 = 3⁴), phir niyam lagao. Minus power dikhe to turant “ulta karo” bolo, minus jawab mat banao. Aur standard form ke baad ek line ka check (wapas aam number banao) zaroor likho. ⚡
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👉 Aage Kya?
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