Indices and surds are two sides of the same coin. Once you see that, the chapter becomes straightforward.
Indices (singular: index) are just another name for exponents or powers. When you write 2⁵, the number 5 is the index. It tells you how many times to multiply 2 by itself. In Indian classrooms you'll often hear the word "घात" (ghaat) for this — it means the same thing.
Surds are roots that cannot be expressed as exact rational numbers. √2 = 1.414... goes on forever without repeating — that makes it a surd. On the other hand, √9 = 3 is a whole number, so it is NOT a surd. The key test: if the root simplifies to a rational number (integer or fraction), it is not a surd.
Here is a useful analogy. Think of an index like a recipe multiplier: 3⁴ means "make this dish (3) four times over and multiply the result." Indices just describe repeated multiplication. Surds, meanwhile, are the opposite operation — you are asking "what number, multiplied by itself a certain number of times, gives me this?" When the answer is messy, you have a surd.
For SSC MTS specifically, the questions stick to:
√2 vs ∛3)√(a + 2√b)You will not see abstract proofs or higher-order surds. What you will see is the comparison trick and the √(a + 2√b) pattern — both appear in recent PYQs. This page covers both in detail.
These six laws cover every index question at this level. Memorise them once and apply mechanically.
| Law | Statement | Example |
|-----|-----------|---------|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ | (2³)⁴ = 2¹² |
| Zero Index | a⁰ = 1 (a ≠ 0) | 999⁰ = 1 |
| Negative Index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional Index | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = ∛(8²) = ∛64 = 4 |
The fractional index law is the bridge between indices and surds: √a = a^(1/2), ∛a = a^(1/3). This is why the two topics are taught together.
√5, ∛7 — a single term√3 + √5 — two surd terms added(√a + √b) and (√a - √b) — their product is rational: (√a + √b)(√a - √b) = a - bThe conjugate idea matters when rationalising denominators. If you see 1/(√5 + √2), multiply top and bottom by (√5 - √2) to clear the surd from the denominator.
This is the single most-tested surd skill in SSC MTS. The method has exactly two steps.
Step 1: Find the LCM of the root orders. Step 2: Convert both surds so they have the same root order, then compare the numbers inside.
Look — √2 vs ∛3:
√2 = 2^(1/2) = 2^(3/6) = ⁶√(2³) = ⁶√8∛3 = 3^(1/3) = 3^(2/6) = ⁶√(3²) = ⁶√9⁶√8 vs ⁶√9. Since 9 > 8, we get ∛3 > √2.That is the complete method. No calculator needed, no approximation guessing.
√(a + 2√b) PatternThis is the nested surd pattern. Questions of the form √(20 + 2√99) appear regularly. The technique is to split a into two parts p and q such that:
p + q = ap × q = bThen: √(a + 2√b) = √p + √q
The logic: (√p + √q)² = p + q + 2√(pq) = a + 2√b. So the square root of the whole expression is √p + √q.
Example: √(20 + 2√99)
p + q = 20 and pq = 991 × 99, 3 × 33, 9 × 119 + 11 = 20. Yes.√(20 + 2√99) = √11 + √9 = √11 + 3The same method works for √(a - 2√b) — the answer becomes √p - √q (larger minus smaller).
Do not skip this. At least one question per paper is a direct evaluation where recognising a perfect square or cube saves 30 seconds.
Perfect squares up to 625: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625
Perfect cubes up to 512: 1, 8, 27, 64, 125, 216, 343, 512
The key ones that appear in surd problems: √144 = 12, √169 = 13, √81 = 9, √196 = 14, ∛64 = 4, ∛125 = 5.
When comparing surds with different root orders (e.g., √2 vs ∛5 vs ⁴√6), raise each surd to the power of the LCM of all orders, then compare the resulting whole numbers under the same root.
For √2 vs ∛3: LCM(2,3) = 6. Convert both to 6th roots: ⁶√8 vs ⁶√9. Compare 8 and 9. Done in 15 seconds. Standard method (decimal approximation): 40+ seconds with risk of rounding error.
For √(a + 2√b), mentally list factor pairs of b and find the pair that adds to a. Write √p + √q directly.
For √(13 + 2√40): factor pairs of 40 — try 8 × 5 = 40, 8 + 5 = 13. Answer: √8 + √5 = 2√2 + √5. Takes 3 steps. Full algebraic expansion and back-substitution: 8–10 steps. Step saving: more than 60%.
Any non-zero base raised to the power 0 equals 1. When an MCQ asks for the value of a complex expression like (√7 - √5)⁰ + (√3)⁰, do not compute √7 - √5. Each zero-power term equals 1. Count the terms, sum them.
(√7 - √5)⁰ + (√3)⁰ = 1 + 1 = 2. Standard method: approximate each root, subtract, recognise it's raised to 0. Shortcut: 2 seconds flat.
When you see 1/(√a + √b) in an option or mid-problem, multiply numerator and denominator by (√a - √b). The denominator becomes the rational number a - b.
1/(√5 + √2) → multiply by (√5 - √2)/(√5 - √2) → (√5 - √2)/(5 - 2) = (√5 - √2)/3. No decimal approximation needed. Standard method (decimal approximation of each surd): error-prone and slow.
Convert a^(p/q) to ⁿ√(aᵐ) immediately: the denominator q is the root order, the numerator p is the power inside.
125^(2/3) → ∛(125²) → ∛(15625). But smarter: first take ∛125 = 5, then square: 5² = 25. Always apply the root before the power when the base is a perfect power — reduces numbers dramatically. 125^(2/3) in 2 steps vs 4 steps the other way.
In the exam hall, classify the surd/indices question in 5 seconds using this decision tree:
Is it a pure index expression (no roots)? Apply the six index laws directly. Look for the same base — combine exponents. If bases differ, try to express them as powers of a common base (e.g., 4 = 2², 8 = 2³).
Does it have a nested surd √(a ± 2√b)? Use the sum-product split. List factor pairs of b, find the pair summing to a, write the answer.
Is it asking to compare surds with different root orders? Use the LCM Bridge — convert to a common root order and compare the radicands.
Is it a direct evaluation √x or ∛x? Check your memorised list of perfect squares and cubes. If it is not a perfect power, try splitting: √(a × b) = √a × √b.
Does the expression mix surds with multiplication? Evaluate each root separately, then multiply. Do not try to combine under one root unless the bases match.
If none of these fit, write out the numerical value of each surd to two decimal places and compare or compute directly.
Why this question: This is the canonical surd-comparison question for SSC. It tests whether you have the LCM Bridge method or are guessing from memory.
Solving path:
√2 has order 2, ∛3 has order 3.√2 = 2^(1/2) = 2^(3/6) = ⁶√(2³) = ⁶√8∛3 = 3^(1/3) = 3^(2/6) = ⁶√(3²) = ⁶√9⁶√8 and ⁶√9: since 9 > 8, ∛3 > √2.Why this question: Tests whether you know your perfect squares up to 169 and whether you apply the order of operations (roots first, then multiply, then add) correctly.
Solving path:
√144 = 12 (recognise: 12² = 144)√81 = 9 (recognise: 9² = 81)√169 = 13 (recognise: 13² = 169)x = 12 × 9 + 13 = 108 + 13 = 1212x = 2 × 121 = 242Note: a common mistake here is adding √81 + √169 before multiplying by √144. Follow BODMAS — multiplication before addition.
Why this question: Tests the nested surd pattern √(a + 2√b) directly. If you know the sum-product split, this takes 20 seconds. Without it, you would need to test each option by squaring — doable but slower.
Solving path:
√(20 + 2√99)p + q = 20 and pq = 99.1 × 99, 3 × 33, 9 × 11.9 + 11 = 20. Match found: p = 11, q = 9.√(20 + 2√99) = √11 + √9 = √11 + 3.√11 + 3.Verification: (√11 + 3)² = 11 + 9 + 6√11 = 20 + 6√11. Wait — let us recheck. 2√99 = 2√(9×11) = 2 × 3 × √11 = 6√11. And (√11 + 3)² = 11 + 6√11 + 9 = 20 + 6√11. Confirmed.
Wrong order of operations with mixed expressions: In a problem like √144 × √81 + √169, students sometimes add √81 + √169 first. Always evaluate roots, then follow BODMAS — multiplication before addition.
Comparing surds using decimal approximation with rounding errors: √2 ≈ 1.41 and ∛3 ≈ 1.44 are close. If you round to one decimal, both become 1.4 and you conclude they are equal. Use the LCM Bridge method — it gives an exact comparison without rounding.
Forgetting that √9 is not a surd: Some students try to apply surd rules to √9 = 3 and get confused. Check first whether the expression simplifies to a rational number. If it does, treat it as a whole number from that point on.
Applying the wrong formula for nested surds with subtraction: √(a - 2√b) = √p - √q where p > q. Students sometimes write √q - √p and get a negative number. Always subtract the smaller from the larger.
Misapplying the power-of-power law: (aᵐ)ⁿ = aᵐⁿ, not aᵐ⁺ⁿ. This is a consistent error. The two exponents multiply, they do not add. The addition law is for aᵐ × aⁿ (same base, multiplication operation).
Not converting to a common base before applying index laws: 2³ × 4² cannot be simplified using the product law directly because the bases differ. Rewrite 4 = 2² first, giving 2³ × 2⁴ = 2⁷. Skipping this step leads to wrong answers.