Surds and Indices for CAT 2026: The 6-Pattern Method
A Quant cheatsheet that fills a genuine gap in CAT preparation content: surds and indices as a standalone topic. It puts the 12 laws of indices in one table, gives the four surds simplification patterns, introduces a 6-pattern recognition method that covers nearly every question, and works through 12 solved TITA-style examples with the cue for each. It closes with three traps that cost easy, no-negative-marking marks.
Surds and Indices for CAT 2026: The 6-Pattern Method
Almost every surds and indices question in CAT collapses into one of six patterns. Learn the six, and a topic that looks like rote memory turns into quick recognition. Surds and indices CAT questions reward this because so many of them are TITA items with no negative marking: spot the pattern, simplify, enter the value, and bank a near-safe mark. The trouble is that most aspirants never see the patterns. They memorise scattered formulas, freeze on an unfamiliar root, and skip questions that were actually the easiest points on the paper.
This cheatsheet fixes that. You will get the 12 laws of indices in one place, the four surds simplification patterns, the six recognition patterns that cover nearly every question, and 12 solved examples with the cue that should fire in your head. Use it with the Quant formulas master list and the CAT exam guide to slot this topic into your wider Quant plan.
Surds and indices CAT questions reduce to six recognition patterns: common-base, prime-base rewrite, substitution, conjugate rationalisation, denesting, and infinite radicals. Master the 12 laws of indices and four surds patterns below, then drill the 12 solved examples until the cue fires automatically. Many of these appear as TITA items, making them some of the safest, fastest marks in the Quant section.
Why surds and indices are high-return in CAT
The 12 laws of indices in one table
The 4 surds simplification patterns
The 6-pattern method: recognise, then solve
Why Surds and Indices Are High-Return in CAT
This topic earns disproportionate marks for the time you invest. The laws are finite, the question types repeat, and a large share appear as TITA, where a wrong answer costs nothing. That combination makes a confident attempt pure upside. Skip a TITA surds question out of doubt and you have left a free mark on the table.
There is a second payoff. Indices feed logarithms, surds feed number system and algebra, and both show up inside larger questions. Getting fluent here lifts your accuracy across several Quant areas rather than just one narrow slice, which is why it belongs early in a CAT preparation plan, not as a last-week add-on.
The 12 Laws of Indices in One Table
These are the only laws you need. Read them once, then test yourself by covering the right side. Every indices question is an application of two or three of them in sequence.
am × an = am+nam ÷ an = am−n(am)n = amn(a×b)n = anbn(a÷b)n = an÷bna0 = 1 (a ≠ 0)a−n = 1÷ana1/n = nth root of aam/n = nth root of amam = an ⇒ m = nam = bm ⇒ a = ba1 = a, 1n = 1Laws 10 and 11 are the workhorses of equation questions: once both sides share a base, you simply equate exponents. Laws 7, 8, and 9 handle negative and fractional powers, which is where careless sign or root errors creep in.
The 4 Surds Simplification Patterns
Surds questions look varied but reduce to four moves. Learn to spot which one a question wants, and the algebra becomes routine.
1. Rationalise with the conjugate
Cue: a root sits in the denominator.
Multiply top and bottom by the conjugate. For 1÷(√a − √b), multiply by (√a + √b) to get (√a + √b)÷(a − b).
2. Denest a nested radical
Cue: the form √(a ± 2√b).
Write it as √x ± √y where x + y = a and x × y = b. Then √(7 + 2√10) = √5 + √2.
3. Compare surds by a common index
Cue: which is larger, different roots.
Raise both to the LCM of their root indices, then compare radicands. To compare the cube root of 3 and √2, use the 6th power: 32 = 9 beats 23 = 8.
4. Solve infinite or repeated radicals
Cue: a root that repeats forever.
Set the whole expression equal to x, then form a quadratic. For √(6 + √(6 + ...)), write x2 = 6 + x and solve to get x = 3.
The 6-Pattern Method: Recognise, Then Solve
Combine the laws and the surds moves and you get six recognition patterns. Before you write anything, ask which pattern the question fits. The answer to that question is most of the solution.
- Common-base consolidation — combine like bases with laws 1 to 3, then equate exponents.
- Prime-base rewrite — express 4, 8, 16, 27, and 81 as powers of a single prime.
- Substitution — when a term like 2x repeats, let it equal t and solve a quadratic.
- Conjugate rationalisation — clear a root from the denominator.
- Denesting — turn √(a ± 2√b) into √x ± √y.
- Infinite radical — set equal to x and build a quadratic.
Write the six pattern names on a single flashcard. For your next 30 practice questions, your first job is not to solve but to label the pattern in five seconds. Recognition speed, not algebra speed, is what separates a fluent attempt from a slow one on these questions.
Want a steady supply of surds and indices questions sorted by these six patterns?
Drill Surds and Indices Sets12 Solved Questions With Recognition Cues
Six indices, six surds. Each shows the cue that should fire, the key step, and the answer. Cover the answer and try the cue first.
2x+3 = 16.Write 16 as 24, equate exponents: x + 3 = 4.
Answer: x = 1
(35 × 32) ÷ 34.Add and subtract exponents: 35+2−4 = 33.
Answer: 27
4x = 8, find x.Both as powers of 2: 22x = 23, so 2x = 3.
Answer: x = 1.5
22x − 5·2x + 4 = 0.Let t = 2x: t2 − 5t + 4 = 0 ⇒ t = 1 or 4.
Answer: x = 0 or x = 2
27−2/3.Cube root of 27 is 3, then 3−2 = 1÷9.
Answer: 1/9
9x = 3x+2, find x.32x = 3x+2, equate exponents: 2x = x + 2.
Answer: x = 2
1÷(√5 − √3).Multiply by (√5 + √3): denominator becomes 5 − 3 = 2.
Answer: (√5 + √3) / 2
√(9 + 2√14).Find x + y = 9 and xy = 14: x = 7, y = 2.
Answer: √7 + √2
6th powers: 42 = 16 versus 33 = 27.
Answer: √3 is larger
√(12 + √(12 + ...)).x2 = 12 + x ⇒ x2 − x − 12 = 0 ⇒ (x−4)(x+3)=0.
Answer: 4
√50 + √8 − √18.As multiples of √2: 5√2 + 2√2 − 3√2.
Answer: 4√2
(√7 + √3)÷(√7 − √3).Multiply by (√7 + √3): (10 + 2√21)÷4.
Answer: (5 + √21) / 2
3 Traps That Cost Easy Marks
Even prepared aspirants drop marks on surds and indices CAT questions for three avoidable reasons. Each has a one-line fix.
Trap 1: Mishandling negative and fractional exponents
A sign slip on a−n or a flipped numerator and denominator on am/n turns a free mark into a wrong one. The fix is to convert to root form first, then evaluate. Build the habit with the speed math guide.
Trap 2: Forcing algebra instead of rewriting the base
Trying to brute-force 4x = 8 without converting to base 2 wastes time and invites error. Always check whether the numbers share a prime base before doing anything else. This single check solves most indices equations instantly.
Trap 3: Skipping TITA surds out of doubt
Because there is no negative marking on TITA, a reasoned attempt is always worth making. Aspirants who skip these forfeit guaranteed marks. Practise enough that recognition is automatic, then commit. Track how this lifts your Quant score on the CAT score predictor.
Cover every answer above and run only the cues. Can you name the pattern for each of the 12 questions in under ten seconds? Can you state all 12 laws of indices from memory? If yes, this topic is exam-ready. If not, that gap is where one focused practice session pays off most.
- Recognise the pattern before you write; the pattern is most of the answer.
- Rewrite numbers as powers of a common prime before solving any equation.
- Substitute a single variable whenever a power term repeats.
- Rationalise with the conjugate the moment a root sits in a denominator.
- Convert fractional and negative powers to root form to avoid sign errors.
- Never skip a TITA surds or indices question; there is no downside to attempting.
Surds and indices are not a memory test. They are a recognition test, and recognition is trainable.
Turn Quant Formulas Into Reliable Marks
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What are surds and indices in CAT Quant?
Indices are powers, governed by laws like a to the m times a to the n equals a to the m plus n. Surds are irrational roots such as the square root of 2 that cannot be written as a clean fraction. In CAT, the two appear together because most questions ask you to simplify expressions, solve power equations, or rationalise roots using a fixed set of laws and patterns.
How important are surds and indices for CAT 2026?
They are high-return because many appear as TITA items with no negative marking, so a confident attempt carries no downside. The laws are finite and the question types are predictable, which means a few focused hours convert directly into reliable marks. Surds and indices also feed logarithms, number system, and algebra, so mastering them lifts performance across several Quant areas at once.
What is the fastest way to solve indices questions in CAT?
Express every term as a power of a common prime base, then use the laws to combine exponents and equate them. Rewrite 4, 8, and 16 as powers of 2 so the equation lives in one base. When a term like 2 to the x repeats, substitute a single variable for it to turn the problem into a quadratic. Recognising the base or the substitution is the entire skill.
How do I simplify surds quickly?
Use four patterns. Rationalise a denominator with the conjugate. Denest the square root of a plus 2 root b into root x plus root y, where x plus y equals a and x times y equals b. Compare surds by raising both to the LCM of their root indices. Solve an infinite nested radical by setting it equal to x and forming a quadratic. Almost every surds question maps to one of these four.
Are surds and indices questions usually TITA in CAT?
A large share appear as TITA, type-in-the-answer questions, because the simplification leads to a single clean value. Since TITA carries no negative marking, these are among the safest marks in Quant. The right approach is to recognise the pattern, simplify with the laws, and enter the value, treating a well-prepared surds or indices question as a near-guaranteed attempt rather than a gamble.
Optima Learn Editorial Team
CAT preparation specialists publishing structured guides on Quant, VARC, DILR, and IIM admissions. We turn each Quant topic into recognition patterns, clean formula sheets, and solved examples calibrated to the 2026 exam. Explore more at our blog or join the CAT 2026 waitlist.
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