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Post Number: 12/76 | Topic: 1.12 Surds & Indices
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🎉 Number System Chapter Complete Hone Wala Hai!
Post Number: 12/76 | Topic: 1.12 Surds & Indices
Previous: 1.1-1.11 Complete ✅
🎉 Number System Chapter Complete Hone Wala Hai!
Introduction: Indices Aur Surds Kya Hain?
Indices (Exponents/Powers) Kya Hai?
🔹 Index (घातांक) = Number jisko kitni baar multiply karna hai
aⁿ = a × a × a × ... (n times)
a = Base (आधार), n = Index/Exponent (घातांक)
aⁿ = a × a × a × ... (n times)
a = Base (आधार), n = Index/Exponent (घातांक)
Examples:
2³ = 2 × 2 × 2 = 8 (Base=2, Index=3)
5⁴ = 5 × 5 × 5 × 5 = 625
10² = 10 × 10 = 100
2³ = 2 × 2 × 2 = 8 (Base=2, Index=3)
5⁴ = 5 × 5 × 5 × 5 = 625
10² = 10 × 10 = 100
Surds (करणी) Kya Hai?
🔹 Surd = Irrational root jo exact decimal form me nahi likha ja sakta
General form: ⁿ√a (n-th root of a)
General form: ⁿ√a (n-th root of a)
Examples of Surds:
√2 = 1.414... (surd hai, irrational)
√3 = 1.732...
∛5 = 1.709...
NOT Surds:
√4 = 2 (perfect square, rational)
√9 = 3 (rational)
∛8 = 2 (rational)
√2 = 1.414... (surd hai, irrational)
√3 = 1.732...
∛5 = 1.709...
NOT Surds:
√4 = 2 (perfect square, rational)
√9 = 3 (rational)
∛8 = 2 (rational)
🎯 Exam Importance: SSC CGL me direct questions aate hain:
- Laws of indices application (बहुत important!)
- Simplification of expressions with powers
- Rationalization of surds
- Comparison of surds
- Simplification of complex expressions
Laws of Indices MUST KNOW
Ye 7 laws bahut important hain. Sabhi yaad karo!
Law 1: Product Law (गुणा नियम)
aᵐ × aⁿ = aᵐ⁺ⁿ
Rule: Same base multiply karte waqt powers ADD karo
Rule: Same base multiply karte waqt powers ADD karo
Example 1: 2³ × 2⁴ = ?
= 2³⁺⁴ = 2⁷ = 128 ✓
Example 2: x⁵ × x³ = x⁵⁺³ = x⁸ ✓
Example 3: 5² × 5 × 5³ = 5²⁺¹⁺³ = 5⁶ ✓
= 2³⁺⁴ = 2⁷ = 128 ✓
Example 2: x⁵ × x³ = x⁵⁺³ = x⁸ ✓
Example 3: 5² × 5 × 5³ = 5²⁺¹⁺³ = 5⁶ ✓
Law 2: Quotient Law (भाग नियम)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Rule: Same base divide karte waqt powers SUBTRACT karo
Rule: Same base divide karte waqt powers SUBTRACT karo
Example 1: 3⁵ ÷ 3² = ?
= 3⁵⁻² = 3³ = 27 ✓
Example 2: x⁷ ÷ x⁴ = x⁷⁻⁴ = x³ ✓
Example 3: 10⁶ ÷ 10² = 10⁴ = 10000 ✓
= 3⁵⁻² = 3³ = 27 ✓
Example 2: x⁷ ÷ x⁴ = x⁷⁻⁴ = x³ ✓
Example 3: 10⁶ ÷ 10² = 10⁴ = 10000 ✓
Law 3: Power of a Power (घात की घात)
(aᵐ)ⁿ = aᵐˣⁿ
Rule: Power ke upar power ho toh MULTIPLY karo
Rule: Power ke upar power ho toh MULTIPLY karo
Example 1: (2³)² = ?
= 2³ˣ² = 2⁶ = 64 ✓
Example 2: (x²)⁵ = x²ˣ⁵ = x¹⁰ ✓
Example 3: (5²)³ = 5⁶ = 15625 ✓
= 2³ˣ² = 2⁶ = 64 ✓
Example 2: (x²)⁵ = x²ˣ⁵ = x¹⁰ ✓
Example 3: (5²)³ = 5⁶ = 15625 ✓
Law 4: Power of a Product
(ab)ⁿ = aⁿ × bⁿ
Rule: Product ka power = Individual powers ka product
Rule: Product ka power = Individual powers ka product
Example 1: (2×3)² = ?
= 2² × 3² = 4 × 9 = 36 ✓
Verification: (6)² = 36 ✓
Example 2: (xy)³ = x³y³ ✓
= 2² × 3² = 4 × 9 = 36 ✓
Verification: (6)² = 36 ✓
Example 2: (xy)³ = x³y³ ✓
Law 5: Power of a Quotient
(a/b)ⁿ = aⁿ/bⁿ
Rule: Fraction ka power = Numerator aur denominator dono ka power
Rule: Fraction ka power = Numerator aur denominator dono ka power
Example: (2/3)³ = ?
= 2³/3³ = 8/27 ✓
= 2³/3³ = 8/27 ✓
Law 6: Zero Power
a⁰ = 1 (a ≠ 0)
Rule: Kisi bhi non-zero number ka power 0 ho toh answer 1
Rule: Kisi bhi non-zero number ka power 0 ho toh answer 1
Examples:
5⁰ = 1 ✓
100⁰ = 1 ✓
(-7)⁰ = 1 ✓
(x²y³)⁰ = 1 ✓
5⁰ = 1 ✓
100⁰ = 1 ✓
(-7)⁰ = 1 ✓
(x²y³)⁰ = 1 ✓
Law 7: Negative Power
a⁻ⁿ = 1/aⁿ
Rule: Negative power = Reciprocal with positive power
Rule: Negative power = Reciprocal with positive power
Example 1: 2⁻³ = ?
= 1/2³ = 1/8 = 0.125 ✓
Example 2: 5⁻² = 1/5² = 1/25 = 0.04 ✓
Example 3: (3/4)⁻² = (4/3)² = 16/9 ✓
= 1/2³ = 1/8 = 0.125 ✓
Example 2: 5⁻² = 1/5² = 1/25 = 0.04 ✓
Example 3: (3/4)⁻² = (4/3)² = 16/9 ✓
Fractional Indices (Roots)
⭐ a^(1/n) = ⁿ√a
⭐ a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
⭐ a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Example 1: 16^(1/2) = ?
= √16 = 4 ✓
Example 2: 8^(1/3) = ?
= ∛8 = 2 ✓
Example 3: 27^(2/3) = ?
= (∛27)² = 3² = 9 ✓
Ya: ∛(27²) = ∛729 = 9 ✓
= √16 = 4 ✓
Example 2: 8^(1/3) = ?
= ∛8 = 2 ✓
Example 3: 27^(2/3) = ?
= (∛27)² = 3² = 9 ✓
Ya: ∛(27²) = ∛729 = 9 ✓
Types of Surds
1. Pure Surd (शुद्ध करणी)
Sirf root ke andar value ho, bahar kuch nahi
Examples: √2, √5, ∛7
2. Mixed Surd (मिश्र करणी)
Root ke bahar bhi number ho
Examples: 2√3, 5√2, 3∛5
3. Like Surds (समान करणी)
Same irrational part wale surds
Examples: 2√3, 5√3, 7√3 (sab ke pass √3)
4. Unlike Surds (असमान करणी)
Different irrational parts
Examples: √2, √3, √5 (sab alag)
5. Conjugate Surds (संयुग्मी करणी)
a + √b aur a - √b (sign ka difference)
Examples:
2 + √3 aur 2 - √3 (conjugates)
5 + √7 aur 5 - √7 (conjugates)
2 + √3 aur 2 - √3 (conjugates)
5 + √7 aur 5 - √7 (conjugates)
Operations With Surds
Addition/Subtraction
Sirf like surds add/subtract kar sakte hain
Example 1: 3√2 + 5√2 = ?
= (3+5)√2 = 8√2 ✓
Example 2: 7√5 - 2√5 = 5√5 ✓
Example 3: 2√3 + 4√2 = ?
Unlike surds, cannot simplify ✓
= (3+5)√2 = 8√2 ✓
Example 2: 7√5 - 2√5 = 5√5 ✓
Example 3: 2√3 + 4√2 = ?
Unlike surds, cannot simplify ✓
Multiplication
√a × √b = √(ab)
a√b × c√d = ac√(bd)
a√b × c√d = ac√(bd)
Example 1: √2 × √3 = ?
= √(2×3) = √6 ✓
Example 2: 2√3 × 5√2 = ?
= (2×5)√(3×2) = 10√6 ✓
Example 3: √5 × √5 = √25 = 5 ✓
= √(2×3) = √6 ✓
Example 2: 2√3 × 5√2 = ?
= (2×5)√(3×2) = 10√6 ✓
Example 3: √5 × √5 = √25 = 5 ✓
Division
√a ÷ √b = √(a/b)
a√b ÷ c√d = (a/c)√(b/d)
a√b ÷ c√d = (a/c)√(b/d)
Example: √18 ÷ √2 = ?
= √(18/2) = √9 = 3 ✓
= √(18/2) = √9 = 3 ✓
Rationalization (परिमेयकरण) VERY IMPORTANT
Rationalization = Denominator se surd hatana
Method: Conjugate se multiply/divide karo
Method: Conjugate se multiply/divide karo
Type 1: Simple Surd in Denominator
Example: 1/√2 ko rationalize karo
Step 1: Numerator aur denominator me √2 multiply karo
= (1 × √2)/(√2 × √2)
= √2/2 ✓
Denominator ab rational hai (2)!
Step 1: Numerator aur denominator me √2 multiply karo
= (1 × √2)/(√2 × √2)
= √2/2 ✓
Denominator ab rational hai (2)!
Example 2: 3/√5 = ?
= (3 × √5)/(√5 × √5)
= 3√5/5 ✓
= (3 × √5)/(√5 × √5)
= 3√5/5 ✓
Type 2: Binomial Surd in Denominator
Conjugate use karo:
a + √b ka conjugate = a - √b
a - √b ka conjugate = a + √b
a + √b ka conjugate = a - √b
a - √b ka conjugate = a + √b
Example: 1/(2 + √3) ko rationalize karo
Step 1: Conjugate (2 - √3) se multiply
= [1 × (2 - √3)] / [(2 + √3)(2 - √3)]
Step 2: Denominator solve (a² - b²)
= (2 - √3) / (4 - 3)
= (2 - √3) / 1
= 2 - √3 ✓
Step 1: Conjugate (2 - √3) se multiply
= [1 × (2 - √3)] / [(2 + √3)(2 - √3)]
Step 2: Denominator solve (a² - b²)
= (2 - √3) / (4 - 3)
= (2 - √3) / 1
= 2 - √3 ✓
Example 2: 5/(3 - √2) = ?
Conjugate: 3 + √2
= [5(3 + √2)] / [(3 - √2)(3 + √2)]
= [5(3 + √2)] / (9 - 2)
= [5(3 + √2)] / 7
= (15 + 5√2) / 7 ✓
Conjugate: 3 + √2
= [5(3 + √2)] / [(3 - √2)(3 + √2)]
= [5(3 + √2)] / (9 - 2)
= [5(3 + √2)] / 7
= (15 + 5√2) / 7 ✓
Simplification of Surds
√(ab) = √a × √b
Perfect square factor nikalo
Perfect square factor nikalo
Example 1: √12 ko simplify karo
√12 = √(4 × 3) = √4 × √3 = 2√3 ✓
Example 2: √75 = ?
= √(25 × 3) = 5√3 ✓
Example 3: √200 = ?
= √(100 × 2) = 10√2 ✓
√12 = √(4 × 3) = √4 × √3 = 2√3 ✓
Example 2: √75 = ?
= √(25 × 3) = 5√3 ✓
Example 3: √200 = ?
= √(100 × 2) = 10√2 ✓
Surds Ko Compare Kaise Karein?
Same root index wale surds: Andar ke values compare karo
Different root index: Convert to same index
Different root index: Convert to same index
Example 1: Kaun bada hai: √3 ya √5?
3 < 5, toh √3 < √5 ✓
Example 2: 2√3 ya 3√2?
(2√3)² = 4 × 3 = 12
(3√2)² = 9 × 2 = 18
18 > 12, toh 3√2 > 2√3 ✓
3 < 5, toh √3 < √5 ✓
Example 2: 2√3 ya 3√2?
(2√3)² = 4 × 3 = 12
(3√2)² = 9 × 2 = 18
18 > 12, toh 3√2 > 2√3 ✓
SSC CGL Level Practice Questions
Q1. (2⁵ × 2³) ÷ 2⁴ = ?
(A) 2⁴
(B) 2⁶
(C) 8
(D) 16
Answer: (D) 16
Solution:
2⁵ × 2³ = 2⁵⁺³ = 2⁸
2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16 ✓
(A) 2⁴
(B) 2⁶
(C) 8
(D) 16
Answer: (D) 16
Solution:
2⁵ × 2³ = 2⁵⁺³ = 2⁸
2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16 ✓
Q2. [(3²)³ × 3⁴] ÷ 3⁸ = ?
(A) 3²
(B) 9
(C) 27
(D) 81
Answer: (A) 3² (or B: 9, same thing)
Solution:
(3²)³ = 3⁶
3⁶ × 3⁴ = 3¹⁰
3¹⁰ ÷ 3⁸ = 3² = 9 ✓
(A) 3²
(B) 9
(C) 27
(D) 81
Answer: (A) 3² (or B: 9, same thing)
Solution:
(3²)³ = 3⁶
3⁶ × 3⁴ = 3¹⁰
3¹⁰ ÷ 3⁸ = 3² = 9 ✓
Q3. √2 + √3 ka rationalizing factor kya hai?
(A) √2 - √3
(B) √2 + √3
(C) Cannot be rationalized
(D) √6
Answer: (A) √2 - √3
Solution:
(√2 + √3)(√2 - √3) = (√2)² - (√3)² = 2 - 3 = -1 (rational) ✓
(A) √2 - √3
(B) √2 + √3
(C) Cannot be rationalized
(D) √6
Answer: (A) √2 - √3
Solution:
(√2 + √3)(√2 - √3) = (√2)² - (√3)² = 2 - 3 = -1 (rational) ✓
Q4. (1/√7 - √6) ko rationalize karo. Answer ka denominator kya hoga?
(A) 1
(B) 13
(C) √42
(D) 7 - 6
Answer: (A) 1
Solution:
Conjugate: √7 + √6
= (√7 + √6) / [(√7 - √6)(√7 + √6)]
= (√7 + √6) / (7 - 6)
= (√7 + √6) / 1 ✓
(A) 1
(B) 13
(C) √42
(D) 7 - 6
Answer: (A) 1
Solution:
Conjugate: √7 + √6
= (√7 + √6) / [(√7 - √6)(√7 + √6)]
= (√7 + √6) / (7 - 6)
= (√7 + √6) / 1 ✓
Q5. 16^(3/4) = ?
(A) 4
(B) 8
(C) 12
(D) 64
Answer: (B) 8
Solution:
16^(3/4) = (⁴√16)³
⁴√16 = 2
2³ = 8 ✓
(A) 4
(B) 8
(C) 12
(D) 64
Answer: (B) 8
Solution:
16^(3/4) = (⁴√16)³
⁴√16 = 2
2³ = 8 ✓
Important Formulas (Yaad Karo!)
Indices:
1. aᵐ × aⁿ = aᵐ⁺ⁿ
2. aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3. (aᵐ)ⁿ = aᵐⁿ
4. (ab)ⁿ = aⁿbⁿ
5. (a/b)ⁿ = aⁿ/bⁿ
6. a⁰ = 1
7. a⁻ⁿ = 1/aⁿ
8. a^(1/n) = ⁿ√a
Surds:
1. √a × √b = √(ab)
2. √a ÷ √b = √(a/b)
3. (√a)² = a
4. (a + √b)(a - √b) = a² - b
5. (√a + √b)(√a - √b) = a - b
1. aᵐ × aⁿ = aᵐ⁺ⁿ
2. aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3. (aᵐ)ⁿ = aᵐⁿ
4. (ab)ⁿ = aⁿbⁿ
5. (a/b)ⁿ = aⁿ/bⁿ
6. a⁰ = 1
7. a⁻ⁿ = 1/aⁿ
8. a^(1/n) = ⁿ√a
Surds:
1. √a × √b = √(ab)
2. √a ÷ √b = √(a/b)
3. (√a)² = a
4. (a + √b)(a - √b) = a² - b
5. (√a + √b)(√a - √b) = a - b
Exam Shortcuts & Tricks
🚀 Trick 1: Negative Power Quick Conversion
2⁻³ = 1/8 (directly याद करो common values)
3⁻² = 1/9
5⁻¹ = 1/5
2⁻³ = 1/8 (directly याद करो common values)
3⁻² = 1/9
5⁻¹ = 1/5
🚀 Trick 2: Fractional Powers
√ = power 1/2
∛ = power 1/3
⁴√ = power 1/4
Instantly convert kar sakte ho!
√ = power 1/2
∛ = power 1/3
⁴√ = power 1/4
Instantly convert kar sakte ho!
🚀 Trick 3: Common Surds Values
√2 ≈ 1.414 (याद करो: "I wish I knew" = 1, 4, 1, 4)
√3 ≈ 1.732
√5 ≈ 2.236
√2 ≈ 1.414 (याद करो: "I wish I knew" = 1, 4, 1, 4)
√3 ≈ 1.732
√5 ≈ 2.236
🚀 Trick 4: Conjugate Pattern
(a + b)(a - b) = a² - b²
Ye rationalization me instant use hota hai!
(a + b)(a - b) = a² - b²
Ye rationalization me instant use hota hai!
Common Mistakes (Ye Galtiyan Mat Karna)
❌ Mistake 1: Powers add karte waqt bases different
2³ × 3² ≠ 6⁵ ✗
Bases same hone chahiye law apply karne ke liye!
2³ × 3² ≠ 6⁵ ✗
Bases same hone chahiye law apply karne ke liye!
❌ Mistake 2: Zero power confusion
0⁰ undefined hai (not 1) ✗
Lekin a⁰ = 1 (jab a ≠ 0) ✓
0⁰ undefined hai (not 1) ✗
Lekin a⁰ = 1 (jab a ≠ 0) ✓
❌ Mistake 3: Surd addition galat
√2 + √3 ≠ √5 ✗
Unlike surds ko add nahi kar sakte as-is!
√2 + √3 ≠ √5 ✗
Unlike surds ko add nahi kar sakte as-is!
❌ Mistake 4: Rationalization me sign forget
Conjugate me sign change karna zaroori hai!
(a + √b) ka conjugate = (a - √b) ✓
Conjugate me sign change karna zaroori hai!
(a + √b) ka conjugate = (a - √b) ✓
Homework Practice (Khud Try Karo)
- (5³ × 5² × 5⁴) ÷ 5⁶ = ?
- √50 ko simplest form me likho
- 1/(3 + √5) ko rationalize karo
- 27^(2/3) ka value kya hai?
- 2√3 × 3√5 = ?
🎉 Number System Chapter Complete! 🎉
🏆 Congratulations!
Tumne Chapter 1: Number System ke sabhi 12 topics complete kar liye!
Topics Covered:
✅ 1.1 Classification of Numbers
✅ 1.2 Divisibility Rules
✅ 1.3 Prime Factorization
✅ 1.4 HCF & LCM
✅ 1.5 Square & Square Root
✅ 1.6 Cube & Cube Root
✅ 1.7 Unit Digit & Last 2 Digits
✅ 1.8 Factorials & Number of Zeros
✅ 1.9 Remainder Theorem
✅ 1.10 Decimal & Fractions
✅ 1.11 Simplification & BODMAS
✅ 1.12 Surds & Indices
Next Chapter Preview: Chapter 2: Arithmetic (20 topics) 🚀
Tumne Chapter 1: Number System ke sabhi 12 topics complete kar liye!
Topics Covered:
✅ 1.1 Classification of Numbers
✅ 1.2 Divisibility Rules
✅ 1.3 Prime Factorization
✅ 1.4 HCF & LCM
✅ 1.5 Square & Square Root
✅ 1.6 Cube & Cube Root
✅ 1.7 Unit Digit & Last 2 Digits
✅ 1.8 Factorials & Number of Zeros
✅ 1.9 Remainder Theorem
✅ 1.10 Decimal & Fractions
✅ 1.11 Simplification & BODMAS
✅ 1.12 Surds & Indices
Next Chapter Preview: Chapter 2: Arithmetic (20 topics) 🚀
Next Chapter Preview
📚 Chapter 2: Arithmetic (20 Topics)
Topics Include:
2.1 Ratio & Proportion
2.2 Percentage
2.3 Profit & Loss
2.4 Discount
2.5 Simple Interest
2.6 Compound Interest
2.7 Time & Work
2.8 Pipes & Cisterns
2.9 Time, Speed & Distance
2.10 Trains
2.11 Boats & Streams
2.12 Average
2.13 Mixture & Alligation
2.14 Partnership
...and more!
🎯 Arithmetic is the HIGHEST weightage chapter in SSC CGL!
Topics Include:
2.1 Ratio & Proportion
2.2 Percentage
2.3 Profit & Loss
2.4 Discount
2.5 Simple Interest
2.6 Compound Interest
2.7 Time & Work
2.8 Pipes & Cisterns
2.9 Time, Speed & Distance
2.10 Trains
2.11 Boats & Streams
2.12 Average
2.13 Mixture & Alligation
2.14 Partnership
...and more!
🎯 Arithmetic is the HIGHEST weightage chapter in SSC CGL!
Conclusion
Surds aur Indices powerful tools hain calculation me. Laws yaad karo aur rationalization practice karo – exam me direct 2-3 marks pakke hain!
Revision Strategy: Number System ke sabhi 12 topics ko ek baar phir se revise karo. Formula sheet bana lo aur daily practice karo. Foundation strong hai toh aage ka sab easy ho jayega!
✅ Post Complete!
Number System Chapter: 12/12 COMPLETE! 🎉
Total Progress: 12/76 posts done (15.8%)
Next: Chapter 2 - Arithmetic (Starting with 2.1 Ratio & Proportion)
Number System Chapter: 12/12 COMPLETE! 🎉
Total Progress: 12/76 posts done (15.8%)
Next: Chapter 2 - Arithmetic (Starting with 2.1 Ratio & Proportion)