Hello friend!! Aaj hum aapko ghaatank aur ghaat (exponents and power) padhane jaa rahe hai. Aaj hum aapko iss lesson ko ekdam basic se padhayenge. Hume puri ummid hai ki humare iss lesson se aapka full concept clear ho jaayega aur aapke jitne bhi doubt hai wo aaj ke iss lesson se dur ho jaayenge. Aaj jo hum padhne aur sikhane waale hai, wo kuch iss prakar se hai-:
➣ Ghaatanko ka pryog kyu karte hai ?
➣ Ghaataankiy rup (Exponential form) kya hota hai?
➣ Ghaataankiy rup ka aadhar (base) aur ghaat (power) kya hota hai?
➣ Ghaatank aur ghaat se related 10 basic question aur unke solution.
Toh chaliye aaj ka lesson start karte hai-:
Koi bhi bahot badi sankhya (greater numbers) ko padhana (read), likhna (write), samjhna aur tulna karna (compare) hume bahot hard lagta hai. Jaise nichey diye examples ko dekhiye-:
Example-1. 100000,00,000000000 aur 10000000000000000000 me kya aap inko aashani se padh sakte hai? Yaa kya aap bata sakte hai ki, kon si sankhya badi (greater) hai?
Example-2. 56700000000000000000 aur 5020000000000000000000 me kya aap bata sakte hai ki, kon si sankhya chhoti (smaller) hai?
Aise hi bahut badi sankhyao ko padhne, likhne, samjhne aur tulna karne me aashani ho, jiske liye hum ghaatanko ka use karte hai. Ghaatanko ka use karke hum badi sankhyao ko short form me likh kar aashani se padh sakte hai, likh sakte hai, tulna etc. kar sakte hai.
Jaise-: 100 = 10×10 = 10²
1,000 = 10×10×10 = 10³
10,000 = 10×10×10×10 = 10⁴
100,000 = 10×10×10×10×10 = 105
10² ko hum 10 ki ghat 2 ya 10 ka varg (square) padhte hai.
Issi prakar se hum nichey diye ghaatankiy rup ko iss prakar se padhte hai-:
10³ = 10 ke ghaat 3 ya 10 ka ghan (cube)
10⁴ = 10 ke ghaat 4 ya 10 ki 4th ghaat (चौथी घात)
105 = 10 ke ghaat 5 ya 10 ki 5th ghaat (पाँचवी घात)
Jaise-: 100 ko hum ghaat ke rup me 100 = 10², likhte hai. Iss 10² ko hi 100 ka ghaatankiy rup kahte hai.
Issi prakar se 1,000 = 10³, 10,000 = 10⁴ aur 100,000 = 105 ko ghaatankiy rup (exponential form) kahte hai.
Toh 10³ me 10 ko hum aadhar kahte hai aur 3 ko hum ghaat kahte hai. Issi prakar se,
10² me aadhar = 10 aur ghaat = 2
10⁴ me aadhar = 10 aur ghaat = 4
105 me aadhar = 10 aur ghaat = 5
Upar ke sabhi example me ek hi aadhar 10 hai. Aayiye hum kuch aise examples dekhte hai jisme aadhar alag-alag ho. Jaise-:
8 = 2×2×2 = 2³
aadhar = 2 aur ghaat = 3
81 = 3×3×3×3 = 3⁴
aadhar = 3 aur ghaat = 4
144 = 12×12 = 12²
aadhar = 12 aur ghaat = 2
-343 = (-7)×(-7)×(-7) = (-7)³
aadhar = -7 aur ghaat = 3
3125 = 5×5×5×5×5 = 55
aadhar = 5 aur ghaat = 5
-59049 = (-9)×(-9)×(-9)×(-9)×(-9) = (-9)5
aadhar = (-9) aur ghaat = 5
Question-1. 3×3×3×3×3×3×3×3 aur 4×4×4×4×4×4×4×4×4×4×4 ko ghaatankiy rup me likhkar aadhar aur ghaat batayiye.
Solution-: 3×3×3×3×3×3×3×3 = 38 me aadhar = 3 aur ghaat = 8 Ans.
Issi prakaar se, 4×4×4×4×4×4×4×4×4×4×4 = 4¹¹, aadhar = 4 aur ghaat = 11 Ans.
Ques-2. Maan gyaat kijiye-
(i). 2³ (ii). (-3)⁴
(iii). (-4)5 (iv). 5²
(v). (-6)²
Solution-: (i). 2³ = 2×2×2 = 8 Ans.
(ii). (-3)⁴ = (-3)×(-3)×(-3)×(-3) = 81 Ans.
(iii). (-4)5 = (-4)×(-4)×(-4)×(-4)×(-4) = -1024 Ans.
(iv). 5² = 5×5 = 25 Ans.
(v). (-6)² = (-6)×(-6) = 36 Ans.
Ques-3. 128 ko ghaatankiy rup me badal kar aadhar aur ghaat batayiye.
Solution-: 128 ko ghaatankiy rup me badalane ke liye hum isske factor nikalenge.
2| 128 ∴128 ka ghaatankiy rup
2| 64 128 = 2×2×2×2×2×2×2
2| 32 = 27 Ans.
2| 16 2⁷me aadhar = 2 aur
2| 8 aur ghaat = 7 Ans.
2| 4
2| 2
| 1
Ques-4. 4096 ko 4 ki ghaat ke rup me likhiye.
Solution-: 4096 ko 4 ki ghaat rup me likhne ke liye hume 4096 ka factor nikalna hoga. Lekin yaha par ye dhyaan dijiye ki, hume 4096 ko 4 ki ghaat ke rup me likhna hai issliye factors me hum kewal 4 hi use karenge. Other koi bhi number hum nhi lenge.
4| 4096 ∴4096 ko 4 ki ghaat rup
4| 1024 4096 = 4×4×4×4×4×4
4| 256 = 46 Ans.
4| 64 46 me aadhar = 4 aur
4| 16 aur ghaat = 6 Ans.
4| 4
| 1
Question-5. Prasaarit rup (expanded form) me likhiye-:
(i). x²y5 (ii). 2²×3⁴
(iii). 6²×4³×2² (iv). a⁴b³c²
Solution-: (i). x²y5= x×x×y×y×y×y×y Ans.
(ii). 2²×3⁴= 2×2×3×3×3×3 Ans.
(iii). 6²×4³×2² = 6×6×4×4×4×2×2 Ans.
(iv). a⁴b³c² = a×a×a×a×b×b×b×c×c Ans.
Question-6. Abhaajya gunankhando (prime factorization) ki ghaat rup me likhiye-:
(i).540 (ii). 1575
(iii). 42875 (iv). 88200
Solution-: (i) 540 ka abhaajya gunankhand hum iss prakar se kar sakte hai-:
2| 540 ∴ 540 = 2×2×3×3×3×5
2| 270 = 2²×3³×5 Ans.
3| 135
3| 45
3| 15
5| 5
| 1
(ii). 1575 ke abhajy gunankhand
3| 1575 ∴ 1575 = 3×3×5×5×7
3| 525 = 3²×5²×7 Ans.
5| 175
5| 35
7| 7
| 1
(iii). 42875 ke abhajy gunankhand
5|42875 ∴42875 = 5×5×5×7×7×7
5| 8575 = 5³×7³ Ans.
5| 1715
7| 343
7| 49
7| 7
| 1
(iv). 42875 ke abhajy gunankhand
2|88200 ∴88200 = 2×2×2×5×5×7×
2|44100 ×7×9
2|22050 = 2³×5²×7²×9
5|11025 Ans.
5|2205
7| 441
7| 63
9| 9
| 1
Question-7. 6⁴ aur 46 me kon chhota (smaller) hai?
Solution-: 6⁴ = 6×6×6×6 = 1296 aur 46 = 4×4×4×4×4×4 = 4096
Toh 1296 < 4096
∴ 6⁴ < 46
Matlab 6⁶, 46 se chhota (smaller) hai. Ans.
Question-8. (-2)7 aur 7² me kon bada (greater) hai?
Solution-: (-2)7= (-2)×(-2)×(-2)×(-2)×(-2)×(-2)×(-2) = -128 aur 7²= 7×7 = 49
Toh 49 > - 128
∴ 7² < (-2)7
Matlab 7², (-2)7 se bada (greater) hai. Ans.
Question-9. Saral (simplify) kijiye-:
(i). 2×3² (ii). 1⁴×4
(iii). 5³×7² (iv). 6²×8³
Solution-: (i). 2×3² = 2×9
= 18 Ans.
(ii). 1⁴×4 = 1×4 = 4 Ans.
(iii). 5³×7² = 125×49 = 6125 Ans.
(iv). 6²×8³ = 36×512 = 18432 Ans.
Question-10. Saral (simplify) kijiye-:
(i). (-3)×4² (ii). 26×(-7)³
(iii). (-1)×(-5)⁴ (iv). (-4)³×(-5)7
Solution-: (i). (-3)×4² = -3×16
= -48 Ans.
(ii). 26×(-7)³ = 64×(-343)
= 21952 Ans.
(iii). (-1)×(-5)⁴ = (-1)×625
= -625 Ans.
(iv) (-4)³ × (-5)7= (-64)×(-3125)
= 200000 Ans.
Ye bhi padhe-:
❒ Percentage Basic concept, Defination and Examples
❒ Whole number defination with example and properties in hindi
❒ Square karne ki shortcut trick up to 100 in hindi
Hume ummid hai ki, aapko aaj ka lesson "ghaatank aur ghaat (exponents and power)" achhe se samjh aaya hoga aur ab aapke sabhi doubt dur ho gaye honge. Lekin phir bhi agar aapko koi question ho ya koi problem ho ya koi doubt ho toh aap hume comment karke bata sakte hai. Yaa aap apna question, doubt ya problem nichey diye contact form se bhi send kar sakte hai.
Aap chahe toh apne question ki image humare Facebook page massage karke bhi send kar sakte hai.
Hum aapke question, problem ya doubt ko solve karne ki puri koshish karenge.
Hum aaj ke lesson ki tarah hi mathematics ke ek-ek lesson ko full details me likhkar aapko padhate hai aur aapki math strong karne ki puri koshish karte hai. Ab aap hume bataye ki, "Next lesson me aap kon-sa math ka topic sikhana chahte hai?"
Hum uss topic ko full details me likhkar aapko padhayenge aur aapko samjhayenge. Toh jaldi se apna topic comment ya massage kare.
At last me agar aapko aaj ka lesson helpful laga ho toh please ek share jarur kare. Aapke ek share karne se hum bahut motivate hote hai aur humari bahut help hoti hai toh please ek share jarur kariye.
➣ Ghaatanko ka pryog kyu karte hai ?
➣ Ghaataankiy rup (Exponential form) kya hota hai?
➣ Ghaataankiy rup ka aadhar (base) aur ghaat (power) kya hota hai?
➣ Ghaatank aur ghaat se related 10 basic question aur unke solution.
Toh chaliye aaj ka lesson start karte hai-:
Exponent and power | Ghaatank aur Ghaat ka Basic Concept Hindi
Toh sabse pahle hum jaan lete hai ki, hum Ghaatanko ka pryog kyu karte hai?Example-1. 100000,00,000000000 aur 10000000000000000000 me kya aap inko aashani se padh sakte hai? Yaa kya aap bata sakte hai ki, kon si sankhya badi (greater) hai?
Example-2. 56700000000000000000 aur 5020000000000000000000 me kya aap bata sakte hai ki, kon si sankhya chhoti (smaller) hai?
Aise hi bahut badi sankhyao ko padhne, likhne, samjhne aur tulna karne me aashani ho, jiske liye hum ghaatanko ka use karte hai. Ghaatanko ka use karke hum badi sankhyao ko short form me likh kar aashani se padh sakte hai, likh sakte hai, tulna etc. kar sakte hai.
Jaise-: 100 = 10×10 = 10²
1,000 = 10×10×10 = 10³
10,000 = 10×10×10×10 = 10⁴
100,000 = 10×10×10×10×10 = 105
10² ko hum 10 ki ghat 2 ya 10 ka varg (square) padhte hai.
Issi prakar se hum nichey diye ghaatankiy rup ko iss prakar se padhte hai-:
10³ = 10 ke ghaat 3 ya 10 ka ghan (cube)
10⁴ = 10 ke ghaat 4 ya 10 ki 4th ghaat (चौथी घात)
105 = 10 ke ghaat 5 ya 10 ki 5th ghaat (पाँचवी घात)
Ghaataankiy rup (Exponential form) kya hota hai?
Kisi bhi bahut badi sankhya (greater number) jisko likhne, padhne tulna karne etc. me problem hoti ho toh aisi sankhya ko short form me "ghaat" ke rup me likhne ko hum "ghaatankiy rup (exponential form)" kahte hai.Jaise-: 100 ko hum ghaat ke rup me 100 = 10², likhte hai. Iss 10² ko hi 100 ka ghaatankiy rup kahte hai.
Issi prakar se 1,000 = 10³, 10,000 = 10⁴ aur 100,000 = 105 ko ghaatankiy rup (exponential form) kahte hai.
Ghaatankiy rup (exponential form) ka aadhar (base) aur ghaat (power)
Agar hum 1000 ko ghaatankiy rup me likhe toh 1000 = 10×10×10 = 10³ likhte hai. Ab 10³ me hume jaanana hai ki, aadhar (base) aur ghaat (power) kya hai?Toh 10³ me 10 ko hum aadhar kahte hai aur 3 ko hum ghaat kahte hai. Issi prakar se,
10² me aadhar = 10 aur ghaat = 2
10⁴ me aadhar = 10 aur ghaat = 4
105 me aadhar = 10 aur ghaat = 5
Upar ke sabhi example me ek hi aadhar 10 hai. Aayiye hum kuch aise examples dekhte hai jisme aadhar alag-alag ho. Jaise-:
8 = 2×2×2 = 2³
aadhar = 2 aur ghaat = 3
81 = 3×3×3×3 = 3⁴
aadhar = 3 aur ghaat = 4
144 = 12×12 = 12²
aadhar = 12 aur ghaat = 2
-343 = (-7)×(-7)×(-7) = (-7)³
aadhar = -7 aur ghaat = 3
3125 = 5×5×5×5×5 = 55
aadhar = 5 aur ghaat = 5
-59049 = (-9)×(-9)×(-9)×(-9)×(-9) = (-9)5
aadhar = (-9) aur ghaat = 5
Ghaatank aur ghaat se related 10 basic question aur unke solution:
Hume ummid hai ki ab aap ghaatank aur ghaat ko achhe se sikh gaye honge. Aayiye inn se related kuch Basic aur Simple question ko solve karna sikhte hai.Question-1. 3×3×3×3×3×3×3×3 aur 4×4×4×4×4×4×4×4×4×4×4 ko ghaatankiy rup me likhkar aadhar aur ghaat batayiye.
Solution-: 3×3×3×3×3×3×3×3 = 38 me aadhar = 3 aur ghaat = 8 Ans.
Issi prakaar se, 4×4×4×4×4×4×4×4×4×4×4 = 4¹¹, aadhar = 4 aur ghaat = 11 Ans.
Ques-2. Maan gyaat kijiye-
(i). 2³ (ii). (-3)⁴
(iii). (-4)5 (iv). 5²
(v). (-6)²
Solution-: (i). 2³ = 2×2×2 = 8 Ans.
(ii). (-3)⁴ = (-3)×(-3)×(-3)×(-3) = 81 Ans.
(iii). (-4)5 = (-4)×(-4)×(-4)×(-4)×(-4) = -1024 Ans.
(iv). 5² = 5×5 = 25 Ans.
(v). (-6)² = (-6)×(-6) = 36 Ans.
Ques-3. 128 ko ghaatankiy rup me badal kar aadhar aur ghaat batayiye.
Solution-: 128 ko ghaatankiy rup me badalane ke liye hum isske factor nikalenge.
2| 128 ∴128 ka ghaatankiy rup
2| 64 128 = 2×2×2×2×2×2×2
2| 32 = 27 Ans.
2| 16 2⁷me aadhar = 2 aur
2| 8 aur ghaat = 7 Ans.
2| 4
2| 2
| 1
Ques-4. 4096 ko 4 ki ghaat ke rup me likhiye.
Solution-: 4096 ko 4 ki ghaat rup me likhne ke liye hume 4096 ka factor nikalna hoga. Lekin yaha par ye dhyaan dijiye ki, hume 4096 ko 4 ki ghaat ke rup me likhna hai issliye factors me hum kewal 4 hi use karenge. Other koi bhi number hum nhi lenge.
4| 4096 ∴4096 ko 4 ki ghaat rup
4| 1024 4096 = 4×4×4×4×4×4
4| 256 = 46 Ans.
4| 64 46 me aadhar = 4 aur
4| 16 aur ghaat = 6 Ans.
4| 4
| 1
Question-5. Prasaarit rup (expanded form) me likhiye-:
(i). x²y5 (ii). 2²×3⁴
(iii). 6²×4³×2² (iv). a⁴b³c²
Solution-: (i). x²y5= x×x×y×y×y×y×y Ans.
(ii). 2²×3⁴= 2×2×3×3×3×3 Ans.
(iii). 6²×4³×2² = 6×6×4×4×4×2×2 Ans.
(iv). a⁴b³c² = a×a×a×a×b×b×b×c×c Ans.
Question-6. Abhaajya gunankhando (prime factorization) ki ghaat rup me likhiye-:
(i).540 (ii). 1575
(iii). 42875 (iv). 88200
Solution-: (i) 540 ka abhaajya gunankhand hum iss prakar se kar sakte hai-:
2| 540 ∴ 540 = 2×2×3×3×3×5
2| 270 = 2²×3³×5 Ans.
3| 135
3| 45
3| 15
5| 5
| 1
(ii). 1575 ke abhajy gunankhand
3| 1575 ∴ 1575 = 3×3×5×5×7
3| 525 = 3²×5²×7 Ans.
5| 175
5| 35
7| 7
| 1
(iii). 42875 ke abhajy gunankhand
5|42875 ∴42875 = 5×5×5×7×7×7
5| 8575 = 5³×7³ Ans.
5| 1715
7| 343
7| 49
7| 7
| 1
(iv). 42875 ke abhajy gunankhand
2|88200 ∴88200 = 2×2×2×5×5×7×
2|44100 ×7×9
2|22050 = 2³×5²×7²×9
5|11025 Ans.
5|2205
7| 441
7| 63
9| 9
| 1
Question-7. 6⁴ aur 46 me kon chhota (smaller) hai?
Solution-: 6⁴ = 6×6×6×6 = 1296 aur 46 = 4×4×4×4×4×4 = 4096
Toh 1296 < 4096
∴ 6⁴ < 46
Matlab 6⁶, 46 se chhota (smaller) hai. Ans.
Question-8. (-2)7 aur 7² me kon bada (greater) hai?
Solution-: (-2)7= (-2)×(-2)×(-2)×(-2)×(-2)×(-2)×(-2) = -128 aur 7²= 7×7 = 49
Toh 49 > - 128
∴ 7² < (-2)7
Matlab 7², (-2)7 se bada (greater) hai. Ans.
Question-9. Saral (simplify) kijiye-:
(i). 2×3² (ii). 1⁴×4
(iii). 5³×7² (iv). 6²×8³
Solution-: (i). 2×3² = 2×9
= 18 Ans.
(ii). 1⁴×4 = 1×4 = 4 Ans.
(iii). 5³×7² = 125×49 = 6125 Ans.
(iv). 6²×8³ = 36×512 = 18432 Ans.
Question-10. Saral (simplify) kijiye-:
(i). (-3)×4² (ii). 26×(-7)³
(iii). (-1)×(-5)⁴ (iv). (-4)³×(-5)7
Solution-: (i). (-3)×4² = -3×16
= -48 Ans.
(ii). 26×(-7)³ = 64×(-343)
= 21952 Ans.
(iii). (-1)×(-5)⁴ = (-1)×625
= -625 Ans.
(iv) (-4)³ × (-5)7= (-64)×(-3125)
= 200000 Ans.
Ye bhi padhe-:
❒ Percentage Basic concept, Defination and Examples
❒ Whole number defination with example and properties in hindi
❒ Square karne ki shortcut trick up to 100 in hindi
Hume ummid hai ki, aapko aaj ka lesson "ghaatank aur ghaat (exponents and power)" achhe se samjh aaya hoga aur ab aapke sabhi doubt dur ho gaye honge. Lekin phir bhi agar aapko koi question ho ya koi problem ho ya koi doubt ho toh aap hume comment karke bata sakte hai. Yaa aap apna question, doubt ya problem nichey diye contact form se bhi send kar sakte hai.
Aap chahe toh apne question ki image humare Facebook page massage karke bhi send kar sakte hai.
Hum aapke question, problem ya doubt ko solve karne ki puri koshish karenge.
Hum aaj ke lesson ki tarah hi mathematics ke ek-ek lesson ko full details me likhkar aapko padhate hai aur aapki math strong karne ki puri koshish karte hai. Ab aap hume bataye ki, "Next lesson me aap kon-sa math ka topic sikhana chahte hai?"
Hum uss topic ko full details me likhkar aapko padhayenge aur aapko samjhayenge. Toh jaldi se apna topic comment ya massage kare.
At last me agar aapko aaj ka lesson helpful laga ho toh please ek share jarur kare. Aapke ek share karne se hum bahut motivate hote hai aur humari bahut help hoti hai toh please ek share jarur kariye.
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