19 Surds and Indices PYQ (Solutions)

Master Surds and Indices for CAT 2026 with practice questions and detailed explanations

CAT 2025

Surds & Indices (irrational roots, exponent rules) are another minor algebra topic. CAT questions on surds/indices are infrequent and usually simple if they appear. They often come intertwined with other problems (like simplifying an expression within a larger algebra question).

The combined category of “Logarithms, Surds and Indices” contributed about 1–2 questions in many years.
- For instance, CAT 2019 had maybe one question using indices (since “Modern Math” was only 1 question total that year).
- CAT 2020 saw up to 4 questions involving logs/surds in one slot (slot 1) but only 1 in another, indicating volatility.
Many papers had 0 direct surd questions.

Summary: Surds/indices are not a guaranteed yearly feature – they appeared sporadically (often bundled with logs). When they do appear, usually 1 question at most in a given slot.

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Weightage Over Past Years

Year	Q.NONumber of questions	Difficulty Level
2024	5	Medium
2023	4	Hard
2022	1	Hard
2021	1	Medium
2020	2	Medium
2019	3	Hard
2018	1	Hard
2017	2	Medium

CAT 2024 Surds and Indices questions

Question 1

Slot-1

If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$ , where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is.

Question 2

Slot-1

The sum of all real values of $k$ for which $\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}$ , is

$\frac{2}{3}$

$-\frac{2}{3}$

$\frac{4}{3}$

$-\frac{4}{3}$

Question 3

Slot-2

If $(x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}$ , then $x$ equals

Question 4

Slot-3

If $(a+b \sqrt{3})^2=52+30 \sqrt{3}$ , where $a$ and $b$ are natural numbers, then $a+b$ equals

Question 5

Slot-3

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$ , is

$2 \log_{10} 2$

$3 \log_{10} 2$

$4 \log_{10} 2$

$\log_{10} 2$

CAT 2023 Surds and Indices questions

Question 1

Slot-1

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

$4 \sqrt{5}$

$2 \sqrt{7}$

$3 \sqrt{31}$

$3 \sqrt{7}$

Question 2

Slot-2

The sum of all possible values of $x$ satisfying the equation $2^{4x^2} - 2^{2x^2 + x + 16} + 2^{2x + 30} = 0$ is

$\frac{3}{2}$

$\frac{5}{2}$

$\frac{1}{2}$

Question 3

Slot-3

Let $n$ and $m$ be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$ , which can be expressed as powers of 2 . Then, the smallest possible value of $n+m$ is

Question 4

Slot-3

If $x$ is a positive real number such that $x^8 + \left(\frac{1}{x}\right)^8 = 47$ , then the value of $x^9 + \left(\frac{1}{x}\right)^9$ is

$40 \sqrt{5}$

$36 \sqrt{5}$

$30 \sqrt{5}$

$34 \sqrt{5}$

CAT 2022 Surds and Indices questions

Question 1

Slot-3

If $\left(\sqrt{\frac{7}{5}}\right)^{3 x-y}=\frac{875}{2401}$ and $\left(\frac{4 a}{b}\right)^{6 x-y}=\left(\frac{2 a}{b}\right)^{y-6 x}$ , for all non-zero real values of $a$ and $b$ , then the value of $x+y$ is

CAT 2021 Surds and Indices questions

Question 1

Slot-3

If $n$ is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999$ , then the smallest value of $n$ is

CAT 2020 Surds and Indices questions

Question 1

Slot-1

If $x = (4096)^{7+4\sqrt{3}}$ , then which of the following equals 64?

$\frac{x^{7/2}}{x^{4/\sqrt{3}}}$

$\frac{x^{7}}{x^{4\sqrt{3}}}$

$\frac{x^{7/2}}{x^{2\sqrt{3}}}$

$\frac{x^{7}}{x^{2\sqrt{3}}}$

Question 2

Slot-3

If a, b, c are non-zero and $14^{a} = 36^{b} = 84^{c}$ , then $6b\left(\frac{1}{c} - \frac{1}{a}\right)$ is equal to

CAT 2019 Surds and Indices questions

Question 1

Slot-1

If $m$ and $n$ are integers such that:
$\frac{2^{19} \cdot 3^4}{4^2 \cdot 9^m \cdot 8^n} = \frac{3^n}{16^m \cdot \sqrt[4]{64}}$
Then what is the value of $m$ ?

-16

-24

-12

-20

Question 2

Slot-2

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

$\frac{\log_{2}3}{3}$

$$ \log_{2} 9 $$

$\frac{\log_{2}7}{3}$

$$ \log_{2} 27 $$

Question 3

Slot-2

Question:

If $5^x - 3^y = 13438$ and $5^{x-1} + 3^{y+1} = 9686$ , then $x+y$ equals [TITA]

CAT 2018 Surds and Indices questions

Question 1

Slot-1

Given that $x^{2018} y^{2017} = 1/2$ and $x^{2016} y^{2019} = 8$ , the value of $x^2 + y^3$ ?

$\frac{37}{4}$

$\frac{31}{4}$

$\frac{35}{4}$

$\frac{33}{4}$

CAT 2017 Surds and Indices questions

Question 1

Slot-1

If $9^{2x-1} - 81^{x-1} = 1944$ , then $x$ is

$\dfrac{9}{4}$

$\dfrac{4}{9}$

$\dfrac{1}{3}$

Question 2

Slot-2

If $9^{x - \frac{1}{2}} - 2^{2x - 2} = 4^{x - 3} \times 2^{2x - 3}$ , then $x$ is

$\frac{3}{2}$

$\frac{2}{5}$

$\frac{3}{4}$

$\frac{4}{9}$

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19 Surds and Indices PYQ (Solutions)

Weightage Over Past Years

If (a+bn)(a+b \sqrt{n})(a+bn​) is the positive square root of (29−125)(29-12 \sqrt{5})(29−125​), where aaa and bbb are integers, and nnn is a natural number, then the maximum possible value of (a+b+n)(a+b+n)(a+b+n) is.

The sum of all real values of kkk for which (18)k×(132768)13=18×(132768)1k\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}(81​)k×(327681​)31​=81​×(327681​)k1​ , is

If (x+62)12−(x−62)12=22(x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}(x+62​)21​−(x−62​)21​=22​ , then xxx equals

If (a+b3)2=52+303(a+b \sqrt{3})^2=52+30 \sqrt{3}(a+b3​)2=52+303​ , where aaa and bbb are natural numbers, then a+ba+ba+b equals

The sum of all distinct real values of xxx that satisfy the equation 10x+410x=81210^x + \frac{4}{10^x} = \frac{81}{2}10x+10x4​=281​, is

If 5x+9+5x−9=3(2+2)\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})5x+9​+5x−9​=3(2+2​) , then 10x+9\sqrt{10 x+9}10x+9​ is equal to

The sum of all possible values of xxx satisfying the equation 24x2−22x2+x+16+22x+30=02^{4x^2} - 2^{2x^2 + x + 16} + 2^{2x + 30} = 024x2−22x2+x+16+22x+30=0 is

Let nnn and mmm be two positive integers such that there are exactly 41 integers greater than 8m8^m8m and less than 8n8^n8n , which can be expressed as powers of 2 . Then, the smallest possible value of n+mn+mn+m is

If xxx is a positive real number such that x8+(1x)8=47x^8 + \left(\frac{1}{x}\right)^8 = 47x8+(x1​)8=47, then the value of x9+(1x)9x^9 + \left(\frac{1}{x}\right)^9x9+(x1​)9 is

If nnn is a positive integer such that (107)(107)2…(107)n>999(\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999(710​)(710​)2…(710​)n>999 , then the smallest value of nnn is

If x=(4096)7+43x = (4096)^{7+4\sqrt{3}}x=(4096)7+43​, then which of the following equals 64?

If a, b, c are non-zero and 14a=36b=84c14^{a} = 36^{b} = 84^{c}14a=36b=84c, then 6b(1c−1a)6b\left(\frac{1}{c} - \frac{1}{a}\right)6b(c1​−a1​) is equal to

If mmm and nnn are integers such that: 219⋅3442⋅9m⋅8n=3n16m⋅644\frac{2^{19} \cdot 3^4}{4^2 \cdot 9^m \cdot 8^n} = \frac{3^n}{16^m \cdot \sqrt[4]{64}}42⋅9m⋅8n219⋅34​=16m⋅464​3n​ Then what is the value of mmm?

The real root of the equation 26x+23x+2−21=02^{6x} + 2^{3x+2} - 21 = 026x+23x+2−21=0 is

Question: If 5x−3y=134385^x - 3^y = 134385x−3y=13438 and 5x−1+3y+1=96865^{x-1} + 3^{y+1} = 96865x−1+3y+1=9686, then x+yx+yx+y equals [TITA]

Given that x2018y2017=1/2x^{2018} y^{2017} = 1/2x2018y2017=1/2 and x2016y2019=8x^{2016} y^{2019} = 8x2016y2019=8, the value of x2+y3x^2 + y^3x2+y3?

If 92x−1−81x−1=19449^{2x-1} - 81^{x-1} = 194492x−1−81x−1=1944, then xxx is

If 9x−12−22x−2=4x−3×22x−39^{x - \frac{1}{2}} - 2^{2x - 2} = 4^{x - 3} \times 2^{2x - 3}9x−21​−22x−2=4x−3×22x−3, then xxx is

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If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$ , where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is.

The sum of all real values of $k$ for which $\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}$ , is

If $(x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}$ , then $x$ equals

If $(a+b \sqrt{3})^2=52+30 \sqrt{3}$ , where $a$ and $b$ are natural numbers, then $a+b$ equals

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$ , is

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

The sum of all possible values of $x$ satisfying the equation $2^{4x^2} - 2^{2x^2 + x + 16} + 2^{2x + 30} = 0$ is

Let $n$ and $m$ be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$ , which can be expressed as powers of 2 . Then, the smallest possible value of $n+m$ is

If $x$ is a positive real number such that $x^8 + \left(\frac{1}{x}\right)^8 = 47$ , then the value of $x^9 + \left(\frac{1}{x}\right)^9$ is

If $n$ is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999$ , then the smallest value of $n$ is

If $x = (4096)^{7+4\sqrt{3}}$ , then which of the following equals 64?

If a, b, c are non-zero and $14^{a} = 36^{b} = 84^{c}$ , then $6b\left(\frac{1}{c} - \frac{1}{a}\right)$ is equal to

If $m$ and $n$ are integers such that:
$\frac{2^{19} \cdot 3^4}{4^2 \cdot 9^m \cdot 8^n} = \frac{3^n}{16^m \cdot \sqrt[4]{64}}$
Then what is the value of $m$ ?

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

Question:

If $5^x - 3^y = 13438$ and $5^{x-1} + 3^{y+1} = 9686$ , then $x+y$ equals [TITA]

Given that $x^{2018} y^{2017} = 1/2$ and $x^{2016} y^{2019} = 8$ , the value of $x^2 + y^3$ ?

If $9^{2x-1} - 81^{x-1} = 1944$ , then $x$ is

If $9^{x - \frac{1}{2}} - 2^{2x - 2} = 4^{x - 3} \times 2^{2x - 3}$ , then $x$ is