Surds & Indices
Surds और Indices (powers)
Learning Objective
Use laws of exponents and simplify square roots confidently.
🎯 Learning Objective
Use laws of exponents and simplify square roots confidently.
💡 Concept
- a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n)
- (a^m)^n = a^(mn); a^0 = 1; a^(−n) = 1/a^n
- √(ab) = √a × √b; √(a/b) = √a/√b
- Simplify surds: √48 = √(16×3) = 4√3
- Rationalise: 1/√a = √a/a; 1/(√a+√b) → multiply by (√a−√b)
🧮 Key Formulas
a^m·a^n = a^(m+n)
>
(a^m)^n = a^(mn)
>
a^0 = 1
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√48 = 4√3
✏️ Easy Example
Q. Simplify: 2³ × 2⁴ ÷ 2⁵
- Add powers: 3+4 = 7 → 2⁷
- Subtract: 7−5 = 2 → 2²
Answer: 4
🇮🇳 Real-Life Example
Sound doubles every speaker you add: 2, 4, 8, 16 — powers of 2 model real growth.
📝 Exam-Level Example
Q. If 2^x = 32, find x.
- 32 = 2⁵
- Same base → x = 5
Answer: 5
📝 Exam-Level Example
Q. Arrange in ascending order: √2, ∛3, ⁶√6
- LCM of root orders (2,3,6) = 6
- √2 = ⁶√8, ∛3 = ⁶√9, ⁶√6 = ⁶√6
- Compare insides: 6 < 8 < 9
Answer: ⁶√6 < √2 < ∛3
🪄 Memory Trick
To compare different roots, raise all to the LCM of the root orders and compare the numbers inside.
⚠️ Common Mistakes
- ❌ a^m × b^m ≠ (ab)^(m+m) — powers add only for the SAME base
- ❌ √(a+b) ≠ √a + √b
🏆 Exam Tips
- ✅ Learn squares till 30 and cubes till 15
- ✅ x^0 = 1 questions are free marks
📌 Summary
- Same base: powers add/subtract
- Pull perfect squares out of roots
- Compare roots via LCM of orders