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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Surds & Indices
CAT'26 QUANT CHEATSHEET
Every surds & indices formula and CAT shortcut you need for CAT 2026 — on one page.
Surds and Indices sits at the algebraic core of CAT quant — exponential equations, digit-cyclicity problems, and gnarly nested radicals all reduce to the same handful of index laws once you know where to look. The habit that saves the most time is recognising the shape: a quadratic disguised as a power, a nested surd that de-nests into two simple terms, or an infinite tower that collapses to one equation in a single variable. This sheet lays out every formula you need for the CAT quant section: the laws of indices, exponential equations, unit-digit cyclicity, surd simplification and comparison, rationalisation, de-nesting, and infinite radicals and power towers, each with a worked example. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-surds-indices-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Surds & Indices: every formula you need
1Laws of Indices
The rules for combining powers, used to simplify and solve.
am·an = am+n | (am)n = amn | a0=1, a−m=1/am
Example: (25·34·62)/(122·23) = 34 = 81.
CAT Insight: 23×32 ≠ 65 — only add exponents when the bases already match.
2Exponential Equations
A variable sits in the exponent; equate bases or take logs.
same base → equate exponents | quadratic in ax → let t=ax
Example: 4x−3·2x+2=0, t=2x → t2−3t+2=0 → x = 0 or 1.
CAT Hack: Reject t ≤ 0 when substituting t = ax, since a power is always positive.
3Cyclicity: Unit Digit
The last digit of a power repeats in a short fixed cycle.
cycle 4: 2,3,7,8 | cycle 2: 4,9 | cycle 1: 0,1,5,6 — reduce n mod 4
Example: 7123: 123 mod 4 = 3 → 3rd term of (7,9,3,1) → ends in 3.
Common Mistake: If n mod 4 = 0, take the 4th digit of the cycle, not the 1st.
4Surds: Simplify & Classify
A surd is an irrational root that won't reduce to a rational.
√a·√b = √(ab) | √a/√b = √(a/b) | ⁿ√a = a1/n
Example: √50 = 5√2; √72 = 6√2; ³√54 = 3³√2.
CAT Insight: Only like surds combine: 3√5+7√5=10√5, but √2 and √3 cannot.
5Surd Operations
Add or subtract only like surds; multiply and divide freely.
p√a·q√b = pq√(ab) | (x+y)2+(x−y)2 = 2(x2+y2)
Example: 3√6·2√15 = 18√10; and 6√20/(3√5) = 4.
CAT Hack: (x+y)2−(x−y)2 = 4xy — conjugate identities beat expanding everything out.
6Comparing Surds
Raise different root orders to a common power, then compare integers.
ᵊ√a vs ⁿ√b: L = lcm(m,n), test aL/m vs bL/n
Example: ³√4 vs √3: L=6 → 16 vs 27 → ³√4 < √3.
CAT Hack: This test is exact every time — no decimals, just small integer powers.
7Rationalization
Clear surds from a denominator using the conjugate.
1/(√a+√b) = (√a−√b)/(a−b)
Example: 1/(√3+√2) = √3 − √2.
CAT Insight: 1/(√n+√(n+1)) = √(n+1)−√n telescopes, so the sum from n=1 to 99 collapses to √100−1 = 9.
8Reciprocal & Conjugate Trick
If x = a+√b with a²−b=1, its reciprocal is the conjugate.
x + 1/x = 2a | x2+1/x2 = (x+1/x)2−2
Example: x=2+√3 → x+1/x=4 → x2+1/x2 = 14.
CAT Hack: The surd cancels in x + 1/x, then any higher power builds up by the same recurrence.
9Radical Equations
A variable under a root: isolate it, square, then verify.
isolate a surd → square both sides → repeat if needed → check every root
Example: √(x+5) = x−1 → x2−3x−4=0 → x=4 or −1; only x=4 checks out.
Common Mistake: Squaring invents fake roots, so always substitute every solution back into the original equation.
10De-nesting Surds
A nested surd often unpacks into a sum of two simpler surds.
√(a+b√c) = √x+√y, where x+y=a, xy=b2c/4
Example: √(7+4√3): x+y=7, xy=12 → x,y=3,4 → 2+√3.
CAT Hack: Works whenever a2−b2c is a perfect square: here 49−48=1.
11Infinite Nested Radicals
A self-similar infinite root collapses to one quadratic.
x = √(a±x) ⇒ x2∓x−a = 0
Example: x=√(6+√(6+…)) → x2−x−6=0 → x = 3.
CAT Insight: For a descending form √(a−…), the quadratic gives a clean integer when a = n(n±1).
12Infinite Power Tower
A self-similar tower of exponents, solved the same way as a nested radical.
xx^… = k ⇒ xk = k ⇒ x = k1/k
Example: √2 raised to itself infinitely → (√2)k=k → k = 2.
CAT Favourite: Set the whole tower equal to k, then solve xk=k. It converges only for x ≤ e1/e ≈ 1.44.
CAT exam shortcuts, traps & revision
13
CAT Exam Shortcuts
- Add exponents only when bases match: am·an=am+n, never across different bases
- Exponential equations: same base → equate; quadratic shape → substitute t=ax
- Unit digit cyclicity: reduce the exponent mod 4 on the base's cycle
- Rationalise with the conjugate: 1/(√a+√b) = (√a−√b)/(a−b)
- De-nest √(a+b√c) by solving x+y=a, xy=b2c/4
- Infinite radicals/towers: set the whole expression equal to k and solve a single equation in k
14
Most Common CAT Traps
- Adding exponents across different bases: 23·32 ≠ 65.
- Taking the 1st term of a cycle when n mod 4 = 0, instead of the 4th.
- Forgetting to check every root of a radical equation against the original (squaring invents fake roots).
- Substituting t=ax and accepting t ≤ 0, when a power is always positive.
- Trying to combine unlike surds directly, such as adding √2 and √3.
15
30-Second Revision Box
- am·an=am+n; (am)n=amn; a0=1
- Unit digit cycles have length 1, 2 or 4; reduce n mod 4
- Only like surds add: 3√5+7√5=10√5
- Rationalise using the conjugate; compare surds by raising to lcm of the root orders
- Radical equations: isolate, square, then verify every root
- Infinite nested radicals and power towers both collapse to one equation in a single variable k
This topic rewards recognising the underlying equation over grinding through algebra — once a nested radical or a power tower is set equal to k, it usually turns into a single line of work. Drill this sheet until the index laws and the de-nesting pattern are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Number System cheat sheet, since unit-digit cyclicity and remainder logic overlap heavily. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
