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Exponents & Surds for CAT: Formulas, Weightage & Question Types

One prime at a time, size comparisons, surds, and substitutions that turn quadratic.

Exponents & Surds formula sheet

Everything CAT needs from Exponents & Surds, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place. a,b>0a, b > 0 unless stated otherwise.

The laws

Multiplyingam⋅an=am+na^m \cdot a^n = a^{m+n}
Dividingaman=am−n\dfrac{a^m}{a^n} = a^{m-n}
Power of a power(am)n=amn(a^m)^n = a^{mn}
Power of a product(ab)n=anbn(ab)^n = a^n b^n
Zeroa0=1a^0 = 1 for a≠0a \neq 0
Negativea−n=1ana^{-n} = \dfrac{1}{a^n}
Fractionalam/n=(an)ma^{m/n} = \left(\sqrt[n]{a}\right)^m

Solving an exponential equation

Same base both sidesequate the exponents — valid when the base is not 00, 11 or −1-1
Different basesprime-factorise both sides, then match each prime separately
a2xa^{2x} and axa^x both presentsubstitute t=axt = a^x and solve the quadratic
Three terms, middle one doubledlook for A2−2AB+B2=(A−B)2A^2 - 2AB + B^2 = (A-B)^2

f(x)g(x)=1f(x)^{g(x)} = 1 has three doors

1f(x)=1f(x) = 1, any exponent
2g(x)=0g(x) = 0, provided f(x)≠0f(x) \neq 0
3f(x)=−1f(x) = -1, provided g(x)g(x) is even

Surds

Productsab=ab\sqrt{a}\sqrt{b} = \sqrt{ab}
Rationalising1a+b=a−ba−b\dfrac{1}{\sqrt{a}+\sqrt{b}} = \dfrac{\sqrt{a}-\sqrt{b}}{a-b}
Squaring a surd pair(a+bn)2=a2+nb2+2abn(a+b\sqrt{n})^2 = a^2 + nb^2 + 2ab\sqrt{n}
Denestingmatch the rational parts and the  \sqrt{\ } parts separately
If A+BA+B is known and A2−B2A^2-B^2 is knownA−B=A2−B2A+BA - B = \dfrac{A^2-B^2}{A+B}, then add to get AA

Climbing the xn+1xnx^n + \frac{1}{x^n} ladder

Up one rung(xn+1xn)2=x2n+1x2n+2\left(x^n + \dfrac{1}{x^n}\right)^2 = x^{2n} + \dfrac{1}{x^{2n}} + 2
Cubesx3+1x3=(x+1x)3−3(x+1x)x^3 + \dfrac{1}{x^3} = \left(x+\dfrac{1}{x}\right)^3 - 3\left(x+\dfrac{1}{x}\right)
Combining two rungs(xa+1xa)(xb+1xb)=xa+b+1xa+b+xa−b+1xa−b\left(x^a + \dfrac{1}{x^a}\right)\left(x^b + \dfrac{1}{x^b}\right) = x^{a+b} + \dfrac{1}{x^{a+b}} + x^{a-b} + \dfrac{1}{x^{a-b}}

Sizes worth knowing — for comparison questions

2102^{10}10241024, so 210≈1032^{10} \approx 10^3
log⁡23\log_2 3≈1.585\approx 1.585
log⁡25\log_2 5≈2.322\approx 2.322
log⁡27\log_2 7≈2.807\approx 2.807

The 6 Exponents & Surds question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine. Beyond these, CAT has asked 3 rarer one-off shapes here.

  1. 01

    One base, or one prime at a time

  2. 02

    How big, or how small

  3. 03

    Substitute, and it is a quadratic

  4. 04

    Surds to be squared or matched

  5. 05

    Something to the power something equals 1

  6. 06

    The xn+1xnx^n + \frac{1}{x^n} ladder

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Exponents & Surds in CAT: quick answers

How many Exponents & Surds questions come in CAT Quant?
About 1.1 questions in a 22-question Quant section, worth roughly 3 marks. Exponents & Surds appeared in 20 of the 24 CAT papers from 2017 to 2025, 29 questions in all, which ranks it 8 of 25 Quant chapters.
Are Exponents & Surds questions in CAT MCQ or TITA?
31% of the Exponents & Surds questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Exponents & Surds questions does CAT ask?
CAT keeps returning to 6 question shapes in Exponents & Surds: One base, or one prime at a time; How big, or how small; Substitute, and it is a quadratic; Surds to be squared or matched; Something to the power something equals 1; The x^n + \frac{1}{x^n} ladder.

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