CAT Quant · Study & Practice

Surds & Indices

AreaAlgebra DifficultyEasy–Moderate CAT weightage1–2 questions directly + woven into Number System, Logarithms and equation solving

Surds and indices are the grammar of algebra: master them and exponential equations, logarithms, geometric progressions, and most "simplify this monster expression" questions stop being scary. An index (exponent) is just shorthand for repeated multiplication, and a surd is an irrational root that refuses to give a clean value — like √2 or ∛5. CAT loves this chapter because it tests whether you can manipulate powers fluently rather than reach for a calculator (which you do not have). The same five laws of indices handle everything from 2^10 × 2^15 to 16^(3/4), and a single trick — rationalizing the denominator — converts ugly fractions like 1/(√7−√3) into something you can actually compute. This chapter builds that fluency in order: the laws of indices, fractional powers and roots, rationalization with conjugates, and the simplification of nested and compound surds. Along the way you will learn to compare surds without decimals, solve a^x = b^y by matching bases, and spot the telescoping patterns that turn a long sum of fractions into a one-line answer. Every section comes with the fastest exam method and the traps that quietly cost marks.

Topics

1 Laws of Indices 4 worked · 5 practice · 8-Q test Start → 2 Fractional Indices 4 worked · 5 practice · 8-Q test Start → 3 Rationalization 4 worked · 5 practice · 8-Q test Start → 4 Radical Simplification 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Force everything onto a common prime base before doing anything — 8, 16, 32, 64 are all powers of 2; 9, 27, 81 are powers of 3.
  • To compare a^m and b^n, take a convenient common root (often the GCD of the exponents) so the comparison reduces to small numbers.
  • Fractional index a^(m/n): take the nth root FIRST (smaller numbers), then raise to the mth power.
  • Rationalize a two-term surd denominator by its conjugate — (√a + √b)(√a − √b) = a − b kills both roots in one step.
  • A telescoping sum of 1/(√n + √(n+1)) collapses to √(last+1) − √(first) after each term becomes √(n+1) − √n.
  • De-nest √(a + 2√b) by finding x + y = a and xy = b; the answer is √x + √y. No 2 in front? Multiply inside by 2/2 first.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Writing a^m × a^n = a^(mn) instead of a^(m+n) — you ADD exponents when multiplying same bases.
  • Reading a^m^n as (a^m)^n; exponentiation is right-associative, so it means a^(m^n).
  • In reverse, dividing instead of rationalizing — adding the percentage-style error of multiplying numerator only, not both parts of the fraction.
  • Treating (a + b)^n as a^n + b^n — there is no distributive law over a sum for exponents.
  • Comparing surds of different orders by their radicands directly; you must first raise both to the LCM of the root indices.

📈 CAT exam insight & PYQ analysis

Surds and indices rarely headline a CAT question on their own; they show up as the algebraic spine of harder problems — solving a^x = b equations by matching bases, simplifying an expression before a logarithm or GP question can proceed, and the recurring telescoping-surd sum that rewards anyone who knows to rationalize. XAT and SNAP are more likely to ask a direct simplification or a "which is greater" surd comparison. Expect Easy–Moderate difficulty in isolation, but the marks go to students who never reach for slow arithmetic: common-base conversion and conjugate rationalization are the two reflexes worth drilling until they are automatic.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

a^m × a^n = ?Tap to reveal
a^(m+n)
a^m ÷ a^n = ?Tap to reveal
a^(m−n)
(a^m)^n = ?Tap to reveal
a^(m×n)
a^0 = ? and a^(−n) = ?Tap to reveal
1; 1/a^n
a^(m/n) means?Tap to reveal
ⁿ√(a^m) = (ⁿ√a)^m
16^(3/4) = ?Tap to reveal
8
27^(−2/3) = ?Tap to reveal
1/9
Conjugate of √7 − √3?Tap to reveal
√7 + √3 (product = 4)
1/(√a + √b) rationalized?Tap to reveal
(√a − √b)/(a − b)
√(7 + 2√12) = ?Tap to reveal
2 + √3
Compare ∛2 and √3 (raise to 6th)?Tap to reveal
√3 > ∛2 (27 > 4)
1/(√1+√2)+…+1/(√(n−1)+√n) telescopes to?Tap to reveal
√n − 1

📌 Quick revision

Indices follow five laws: add exponents when multiplying same bases, subtract when dividing, multiply for a power of a power, and a^0 = 1 with a^(−n) = 1/a^n. A fractional index a^(m/n) is the nth root raised to the mth power — take the root first. To compare powers or surds, force a common base or raise to the LCM of the root indices. Rationalize a single surd by itself and a two-term denominator by its conjugate, since (√a + √b)(√a − √b) = a − b. De-nest √(a + 2√b) as √x + √y where x + y = a and xy = b. Watch the traps: never add bases, never split (a+b)^n, and remember exponentiation is right-associative.

Chapter test

📝 Surds & Indices — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Surds & Indices when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards