Quick Concept Summary
Indices (Exponents): aⁿ means 'a multiplied by itself n times'. The number a is the base and n is the index (or exponent or power).
Surds: Irrational numbers expressed with root symbols that cannot be simplified to a rational number. √2 = 1.414... is a surd. √4 = 2 is NOT a surd (it simplifies to a rational number).
Connecting link: a^(m/n) = ⁿ√(aᵐ) — fractional powers are roots.
Laws of Indices — All 7
| Law | Formula | Example |
|---|---|---|
| 1. Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| 2. Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| 3. Power of power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| 4. Product power | (ab)ⁿ = aⁿbⁿ | (2×3)² = 4×9 = 36 |
| 5. Zero index | a⁰ = 1 (a≠0) | 7⁰ = 1 |
| 6. Negative index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| 7. Fractional index | a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |
Additional: a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Surds — Types and Rules
Order of a surd: The index of the root. √a is order 2; ∛a is order 3.
Like surds: Surds with the same order and radicand. 3√2 and 5√2 are like surds.
Unlike surds: 3√2 and 3√3 are unlike (different radicand).
Operations:
- Like surds can be added/subtracted: 3√2 + 5√2 = 8√2
- Unlike surds CANNOT be added directly: √2 + √3 ≠ √5
Rationalisation:
| Form | Multiply by | Result |
|---|---|---|
| 1/√a | √a/√a | √a/a |
| 1/(√a + √b) | (√a−√b)/(√a−√b) | (√a−√b)/(a−b) |
| 1/(√a − √b) | (√a+√b)/(√a+√b) | (√a+√b)/(a−b) |
Worked Examples
Example 1 — Laws of Indices (Basic)
Q: Simplify: (2⁴ × 2⁻² × 2⁶) / 2⁵
Numerator: 2⁴ × 2⁻² × 2⁶ = 2^(4−2+6) = 2⁸
Example 2 — Fractional Indices
Q: Evaluate: 27^(2/3) + 16^(3/4)
27^(2/3) = (∛27)² = 3² = 9
16^(3/4) = (⁴√16)³ = 2³ = 8
Result = 9 + 8 = 17
Example 3 — Negative Indices
Q: Simplify: (4/9)^(−1/2)
(4/9)^(−1/2) = (9/4)^(1/2) = √(9/4) = 3/2 = 1.5
Example 4 — Simplifying Surds
Q: Simplify: √(72) − 2√(32) + 3√(18)
√72 = √(36×2) = 6√2
√32 = √(16×2) = 4√2
√18 = √(9×2) = 3√2
Result = 6√2 − 2(4√2) + 3(3√2) = 6√2 − 8√2 + 9√2 = 7√2
Example 5 — Rationalisation
Q: Rationalise the denominator: 5 / (√7 − √2)
Multiply by conjugate (√7 + √2)/(√7 + √2):
= 5(√7 + √2) / (7 − 2) = 5(√7 + √2) / 5 = √7 + √2
Example 6 — Comparing Surds
Q: Which is greater: √3 or ∛4?
Convert to 6th roots (LCM of 2 and 3):
√3 = 3^(1/2) = 3^(3/6) = ⁶√(3³) = ⁶√27
∛4 = 4^(1/3) = 4^(2/6) = ⁶√(4²) = ⁶√16
Since ⁶√27 > ⁶√16, √3 > ∛4
Shortcuts and Tricks
Trick 1 — Simplify before computing
Always convert to the same base where possible. 8 = 2³, 4 = 2², 16 = 2⁴ — all powers of 2. This is the most common trick in indices problems.
Trick 2 — Fractional powers = roots
- n^(1/2) = √n
- n^(1/3) = ∛n
- n^(2/3) = (∛n)²
- n^(3/2) = (√n)³
Trick 3 — Quick rationalisation check
(√a + √b)(√a − √b) = a − b
This is the conjugate product formula. Use it for all 1/(√a ± √b) problems.
Trick 4 — ax = ay implies x = y (if a ≠ 0, 1, −1)
If 2ˣ = 2⁵, then x = 5. Match bases, equate exponents.
TNPSC / Placement Exam Pattern
TNPSC / SSC: 1–2 questions per paper. Usually simplify a complex expression using laws of indices, or compute a fractional power. TNPSC tends to keep it to integer and simple fractional indices.
Bank PO: May include negative indices and rationalisation.
Common types: 1. Simplify using product/quotient law (same base) 2. Evaluate fractional powers (e.g., 125^(2/3)) 3. Simplify surds (like surds addition) 4. Rationalise the denominator 5. Comparing two surds
Practice Questions
| # | Question | Answer |
|---|---|---|
| 1 | 3⁴ × 3² ÷ 3⁵ = ? | 3¹ = 3 |
| 2 | 64^(1/6) = ? | 2 |
| 3 | (125)^(2/3) = ? | 25 |
| 4 | Simplify: √(50) + √(32) | 9√2 |
| 5 | (2/3)^(−2) = ? | 9/4 |
| 6 | Rationalise: 1/(√5 + √3) | (√5−√3)/2 |
| 7 | 5^(x) = 625. Find x. | 4 |
| 8 | Simplify: ⁴√(16a⁸b⁴) | 2a²b |
| 9 | Which is greater: ⁴√5 or ⁶√7? | ⁴√5 (= ¹²√125 vs ¹²√49) |
| 10 | (0.001)^(1/3) = ? | 0.1 |