Hello friend !! Aaj hum aapko Rules of exponent matlab ghaatanko ke niyam aur inn niyamo se related question ko solve karna sikhane waale hai. Aaj hum aapko full details me isske ek-ek rules ko achhe se samjhayenge aur har rules se related question ko bhi solve karna sikhaayenge.

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❒ Exponents and power | Ghaatank aur ghaat Basic concept Hindi

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Exponent Rule

Exponent Rule with 10+ Example Question - Solution

Ghaatanko ke bade - bade question ko aashani se solve karne ke liye kuch niyam banaye gaye. Inn niyamo ki help se question ko solve karna easy ho jaata hai. Ye niyam iss prakaar se hai-:

Rule-1. Ek hi aadhar waali ghaato ka multiply

Rule-2. Ek hi aadhar waali ghaato ka divide

Rule-3. Samaan Ghaatanko waali ghaato ka multiply

Rule-4. Samaan Ghaatanko waali ghaato ka divide
Rule-5. Ek hi ghaat ki ghaat ka niyam
Rule-6. Zero ghaatank waali sankhya

Ye ghaatanko ke basic rules hai. Inke alawa aur bhi rules hote hai. Jo advance hote hai. Unn rule ko hum next topic me sikhenge. Aaj sirf hum inn 6 important basic rules ko sikhenge aur inki help se question ko solve bhi karenge. Toh chaliye inn niyamm ko ek-ek karke sikh lete hai.

Rule-1. Ek hi aadhar waali ghaato ka multiply

Iss niyam ka use hum tab karte hai, "jab two yaa two se adhik samaan aadhar (equal base) ke ghaatankiy sankhyao ka multiply karna ho."

Iss niyam ke anusaar, "jab two ya two se adhik ghaatankiy sankhayo ka jinka adhar samaan (equal) ho toh inka gudanphal (multiplication) ke liye hume kewal innke ghaato ko jod (Add) kar dena hai."

Formula-: iss rule ka general formula iss prakar se hai:

a^m×aⁿ = a^m+n

Yaha par a = Integer (a ≠ 0)
m, n = Whole number

Jaise -: 2⁴×2⁶ ka maan kya hoga?
Solution-: 2⁴×2⁶ = 2⁴⁺⁶
= 2¹⁰ Ans.
Proof -: 2⁴×2⁶= (2×2×2×2)×(2×2×2×2×2×2)
= 2×2×2×2×2×2×2×2×2×2
= 2¹⁰ Ans.

Rule-2. Ek hi aadhar waali ghaato ka Divide

Jab two samaan (equal) ghaat waali sankhyao ko divide karna ho toh uske liye hum iss rule ka use karte hai.

Iss rule ke anusar, divide ka maan nikalne ke liye hume kewal first number ke ghaat me se Second number ke ghaat ko subtract kar dena hai.

Formula-: Iss rule ka general formula iss prakar se hai-:
    a^m÷aⁿ = a^m-n

Jaise-: 5⁶÷5³ = 5^6-3
= 5³ Ans.

Proof-: 5⁶÷5³ =  5×5×5×5×5×5
5×5×5
= 5×5×5
= 5³ Ans.

Rule-3. Samaan ghaatanko waali ghaato ka multiply

Jab two ya two se adhik samaan (equal) ghaatanko waali sankhyao ka multiply karna ho to iss ke liye hum iss niyam ka use karte hai.

Iss niyam ke anusaar, yadi two ya two se adhik samaan ghaat waali sankhya ka multiply karna ho to isska maan gyaat karne ke liye kewal aadhar ka multiply karke samman ghaat laga dete hai.

Formula-: Iss rule ka general formula iss prakar se hai-:
a^m×b^m= (a×b)^m

Jaise-: 2³×4³ = (2×4)³
= 8³ Ans.

Proof-: 2³ ×4³ = (2×2×2)×(4×4×4)
= 2×4×2×4×2×4
= 8×8×8
= 8³ Ans.

Rule-4. Samaan ghaatanko waali ghaato ka divide

Jab two samaan (equal) ghaatanko waali sankhyo ko divide karna ho to iss ke liye hum iss niyam ka use karte hai.

Iss niyam ke anusaar, yadi samaan ghaat waali sankhyao ka divide karna ho to isska maan gyaat karne ke liye kewal aadhar ka divide karke samaan ghaat laga dete hai.

Formula-: Iss rule ka general formula iss prakar se hai-:
    a^m÷b^m= (a÷b)^m

Jaise-: 8⁴÷4⁴ = (8÷4)⁴
= 2⁴ Ans.

Proof-: 8⁴÷4⁴ = 8×8×8×8
4×4×4×4
= 2×2×2×2
= 2⁴ Ans.

Rule-5. Ek hi ghaat ki ghaat ka niyam

Jab kisi number ki ghaat ke upar bhi ghaat laga ho. Jaise-: (3²)⁴ka maan hum iss niyam se gyaat karte hai.
Iss niyam ke anusaar, agar kisi number ki ghaat ke upar bhi ghaat ho to iss ka maan gyaat karne ke liye sabhi ghaato ka multiply kar dete hai.

Formula-: Iss rule ka general formula iss prakar se hai-:
(a^m)ⁿ= a^m×n

Jaise-: (3²)⁴ = 3^2×4
= 3⁸ Ans.

Rule-6. Zero ghaatank waali sankhya

Jab kisi bhi number (jo zero naa ho) ki ghaat zero "0" ho to uska maan humesha 1 hota hai.

Formula-: Iss rule ka general formula iss prakar se hai-:
         a⁰= 1

Jaise-: 2⁰= 1

Proof-: Isske liye hum ghaatanko ke divide waale rule ka use karenge.

⇒   2⁴   =  2×2×2×2  = 1
2⁴    2×2×2×2

L.H.S. me Rule-2 ka use karne par,
⇒ (2⁴÷2⁴) = 1
⇒ 2^4-4= 1
⇒ 2⁰ = 1    Ans.

Exponent Rule se related 10+ Question with Solution (Ghaatanko ke niyam par aadharit 10+ Question aur unke Solution)

Hume ummid hai ki, ab aapko sabhi niyam achhe se samjh aa gaye honge. Ab hum aapko inn niyamo se related question ko solve karna sikhane jaa rahe hai.

Question-1. Saral karke ghaatankiy rup me likhiye-:
(i). 3¹×3⁴×3⁵
(ii). (-2)¹×(-2)²×(-2)³×(-2)⁴

Solution-: (i). 3¹×3⁴×3⁵= 3¹⁺⁴⁺⁵
= 3¹⁰ Ans.

(ii). (-2)¹×(-2)²×(-2)³×(-2)⁴
= (-2)^1+2+3+4
= (-2)¹⁰   Ans.

Question-2. Saral karke ghaatankiy rup me likhiye-:
(i). 5³÷5²
(ii). (-3)⁶÷(-3)⁴
Solution-: (i). 5³÷5²= 5^3-2
= 5¹
= 5    Ans.

(ii). (-3)⁶÷(-3)⁴ = (-3)^6-4
= (-3)² Ans.

Question-3. Saral karke ghaatankiy rup me likhiye-:
(i). 7²×3²×5²
(ii). (-2)³×1³×(-3)³
Solution-: (i). 7²×3²×5² = (7×3×5)²
= (105)² Ans.
(ii). (-2)³×1³×(-3)³= {(-2)×1×(-3)}³
= 6³ Ans.

Question-4. Saral kijiye-:
(i). 8⁵÷4⁵
(ii). (-4)³÷(-3)³

Solution-:(i). 8⁵÷4⁵ = (8÷4)⁵
= 2⁵
= 32   Ans.

(ii). (-4)³÷(-3)³= {(-4)÷(-3)}³
= (4/3)³
= 4³/3³
= 64/27   Ans.

Question-5. Saral kijiye-:
(i). (6³)⁴÷6⁴
(ii). (4⁵)⁴

Solution-: (i). (6³)⁴÷6⁴= 6^3×4÷6⁴
= 6¹²÷6⁴
= 6^12-4
= 6⁸ Ans.
(ii). (4⁵)⁴= 4^5×4
= 4^20. Ans.

Question-6. Maan gyaat kijiye-:
(i). 27²÷ 3²
(ii) 2×5²÷5×2²

Solution-: (i). 27²÷3² = (3³)²÷3²
= 3^3×2÷3²
= 3⁶÷3²
= 3^6-2
= 3⁴    Ans.

(ii). 2×5²÷2²×5 = 2×25÷4×5
= 50÷20
= 5/2    Ans.

Question-7. Maan gyaat kijiye-:
(i). (3²⁰÷3¹⁵)×3³
(ii). [(4³)⁴×4⁵]÷4⁹

Solution-: (i). (3²⁰÷3¹⁵)×3³
= (3^20-¹⁵)×3³

= 3⁵×3³
= 3⁵⁺³
= 3⁸ Ans.

(ii). [(4³)⁴×4⁵]÷4⁹= [(4³)⁴×4⁵]÷4⁹
= [4³^×4×4⁵]÷4⁹
= [4¹²×4⁵]÷4⁹
= [4¹²⁺⁵]÷4⁹
= 4¹⁷÷4⁹
= 4^17-9
= 4⁸ Ans.

Question-8. Maan gyaat kijiye-:
(i). 1⁰+2⁰+3⁰
(ii). 3⁰×4⁰×5⁰
(iii). (2⁰-2⁰)×2⁰
(iv). (5⁰+4⁰)×3⁰

Solution-: (i). 1⁰+2⁰+3⁰ = 1+1+1
= 3    Ans.
(ii). 3⁰×4⁰×5⁰= 1×1×1
= 1    Ans.

(iii). (2⁰-2⁰)×2⁰ = (1-1)×1
= 0×1
= 0 Ans.

(iv). (5⁰+4⁰)×3⁰ = (1+1)×1
= 2×1
= 2    Ans.

Question-9.    5²×7³×3
7×625
ka maan gyaat kijiye.

Solution-:    5²×7³×3
7×625

=    5²×7³×3
7×5⁴

=    5²  ×   7³   × 3
  5⁴7

= 5^2-4× 7^3-1× 3
= 5^-2×7²×3
= (1/5²)×7²×3
= (1/25)×49×3
= 147/25     Ans.

Question-10.   2³×4×t⁴
t³×8×4²
ka maan gyaat kijiye.

Solution-:    2³×4×t⁴
t³×8×4²
=    2³×4×t⁴
t³×2³×4²
=    2³  ×   4 ×   t⁴
2³4²t³

Ab Rule-2 ka use karne par:

= 2^3-3× 4^1-2×t^4-3
= 2⁰× 4^-1× t¹
= 1 × (1/4)× t
= t/4.    Ans.

Ye bhi sikhe-:

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