CAT Surds and Indices Questions 2026 (Concepts, Shortcuts + Practice Set)

Overview: CAT Surds and Indices Questions are among the most scoring yet underestimated topics in CAT Quantitative Ability. They usually appear within Algebra or Number Systems and test whether you can simplify expressions, identify patterns, and apply exponent rules under time pressure. While the formulas are basic, CAT exam pattern often frames them through unfamiliar-looking expressions, making conceptual clarity more valuable than memorisation.

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If you are preparing for CAT exam 2026, mastering this topic can help you pick up quick and accurate marks, especially in questions involving roots, fractional powers, logarithms, and algebraic simplification.

What Are Surds and Indices in CAT 2026?

Indices are powers or exponents that show how many times a number is multiplied by itself. For example:

2⁴ = 2 × 2 × 2 × 2 = 16

Surds are irrational roots that cannot be simplified into rational numbers. Examples include √2, √3, and ∛5.

However, not every root is a surd. For example, √49 = 7. Since the answer is rational, it is not considered a surd.

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CAT Surds and Indices Questions

Practice these CAT-level difficult Surds and Indices questions to strengthen your Quantitative Aptitude preparation.

Q1. If x = √5 + √3, then the value of x⁴ + 1/x⁴ is:

• A. 384

• B. 386

• C. 388

• D. 390

Answer: B

Q2. Simplify: (√48 + √75 − √27) / √3

• A. 4

• B. 5

• C. 6

• D. 7

Answer: C

Q3. If √(x + 9) − √x = 3, then x is:

• A. 0

• B. 4

• C. 9

• D. 16

Answer: A

Q4. Find the value of (√(7 + 4√3) + √(7 − 4√3))²

• A. 12

• B. 14

• C. 16

• D. 18

Answer: C

Q5. If a = √2 + √3, then a⁶ + 1/a⁶ is:

• A. 970

• B. 980

• C. 990

• D. 1000

Answer: B

Q6. Simplify: 1/(√5 − √2) − 1/(√5 + √2)

• A. 2√2/3

• B. 2√5/3

• C. √10/3

• D. 5√2/3

Answer: A

Q7. If √(x + 4) + √(x − 5) = 5, then x is:

• A. 8

• B. 9

• C. 12

• D. 14

Answer: C

Q8. Find the value of ∛(2 + √5) × ∛(2 − √5)

• A. −1

• B. 0

• C. 1

• D. √5

Answer: A

Q9. If 2^x = 8^{x − 2}, then x is:

• A. 2

• B. 3

• C. 4

• D. 5

Answer: B

Q10. Solve for x: 9^{x + 1} = 27^{2x − 3}

• A. 2

• B. 3

• C. 4

• D. 5

Answer: A

Q11. If 3^{2x − 1} = 81^{x − 2}, then x is:

• A. 3

• B. 4

• C. 5

• D. 6

Answer: D

Q12. Find the value of (16/81)^−3/4

• A. 27/8

• B. 81/16

• C. 27/4

• D. 81/8

Answer: C

Q13. Simplify: [27^2/3 × 16^3/4] / 8^4/3

• A. 6

• B. 9

• C. 12

• D. 18

Answer: B

Q14. If 5^{x + 2} = 125^{x − 1}, then x is:

• A. 1

• B. 2

• C. 3

• D. 4

Answer: B

Q15. Find the value of (2¹⁰ + 2⁹ + 2⁸) / 2⁸

• A. 5

• B. 6

• C. 7

• D. 8

Answer: C

Q16. If √a + √b = 7 and √a − √b = 3, then a + b is:

• A. 25

• B. 29

• C. 31

• D. 35

Answer: B

Q17. Simplify: √(50 + 20√6)

• A. 5 + √6

• B. 2√6 + 5

• C. 3√2 + 2√3

• D. 5√2 + 2√3

Answer: D

Q18. Find the value of (√8 + √18) / √2

• A. 3

• B. 4

• C. 5

• D. 6

Answer: C

Q19. If x^2/3 = 16, then the positive value of x is:

• A. 32

• B. 48

• C. 64

• D. 128

Answer: C

Q20. Solve for x: (1/4)^{x − 1} = 8^{2x − 3}

• A. 1

• B. 7/8

• C. 9/8

• D. 5/4

Answer: C

Q21. If √x + 1/√x = 5, then x + 1/x is:

• A. 21

• B. 23

• C. 25

• D. 27

Answer: B

Q22. Find the value of (√2 + √3)⁴ − (√3 − √2)⁴

• A. 20√6

• B. 24√6

• C. 28√6

• D. 30√6

Answer: A

Q23. Simplify: √3/(√3 − 1) + √3/(√3 + 1)

• A. 2

• B. 3

• C. 4

• D. 5

Answer: B

Q24. If 4^x + 4^x + 4^x = 192, then x is:

• A. 2

• B. 3

• C. 4

• D. 5

Answer: B

Q25. Find the value of [81^3/4 − 16^5/4] / √49

• A. 5

• B. 7

• C. 9

• D. 11

Answer: B

Why Are CAT Surds and Indices Questions Important?

CAT questions from CAT Surds and Indices Questions are rarely direct formula-based questions. Instead, they test your ability to manipulate expressions efficiently and avoid unnecessary calculations.

A typical CAT-level question may combine:

• Fractional indices
• Negative powers
• Nested radicals
• Rationalisation
• Exponent comparison
• Algebraic identities
• Approximation and inequalities

This topic is especially useful because it overlaps with Algebra, Logarithms, Quadratic Equations, and Arithmetic calculations.

Essential Laws of Indices for CAT

• a^m × aⁿ = a^m+n
• a^m / aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• a⁰ = 1
• a^-m = 1 / a^m
• a^m/n = ⁿ√(a^m)
• ⁿ√(a^m) = a^m/n

The most important conversion for CAT is:

ⁿ√(a^m) = a^m/n

It helps you turn complicated radicals into manageable exponent expressions.

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Key Surd Concepts You Must Know

Simplifying Surds

Always look for perfect-square or perfect-cube factors inside a radical.

√72 = √(36 × 2) = 6√2

Similarly, ∛54 = ∛(27 × 2) = 3∛2

Rationalisation of Denominators

CAT frequently tests whether you can remove a surd from the denominator.

1 / √5 = √5 / 5

For expressions involving two terms, use the conjugate. For example:

1 / (√3 + 2) = (√3 - 2) / [(√3 + 2)(√3 - 2)] = 2 - √3

Conjugates

For an expression of the form a + b, its conjugate is a - b.

(a + b)(a - b) = a² - b²

This identity helps simplify expressions involving square roots without expanding everything.

CAT Surds and Indices Questions: Common Question Types

1. Simplification-Based Questions

These questions test exponent laws and basic radical simplification.

Example: Simplify:

[16^3/4 × 8^2/3] / 2⁵

Solution:

16^3/4 = (2⁴)^3/4 = 2³ = 8

8^2/3 = (2³)^2/3 = 2² = 4

(8 × 4) / 32 = 1

2. Comparison Questions

These questions require you to compare values without calculating decimals.

Example: Compare 2^1/3, 3^1/4, and 4^1/5.

Instead of approximating, raise each term to the LCM of the denominators. This preserves the order because all terms are positive.

3. Rationalisation Questions

Example: Find the value of:

1 / (√5 - √3)

Solution:

[1 / (√5 - √3)] × [(√5 + √3) / (√5 + √3)] = (√5 + √3) / 2

4. Nested Radical Questions

Example: Simplify:

√(5 + 2√6)

Assume √(5 + 2√6) = √a + √b.

On squaring both sides:

a + b + 2√ab = 5 + 2√6

Therefore, a + b = 5 and ab = 6. Hence, a = 2 and b = 3.

So, √(5 + 2√6) = √2 + √3.

5. Equation-Based Questions

Example: If x^1/2 + x^-1/2 = 3, find x + 1/x.

Squaring both sides:

x + 1/x + 2 = 9

Therefore, x + 1/x = 7.

Advanced Shortcuts In CAT Surds and Indices Questions

Convert Everything to a Common Base

Whenever possible, express numbers using prime bases.

For example:

27^2/3 × 9^1/2 = (3³)^2/3 × (3²)^1/2 = 3² × 3 = 27

Do Not Expand Too Early

Expressions such as (√7 + √5)² should be expanded only when required. In many CAT questions, recognising an identity or using a conjugate is enough.

Use Approximation Carefully

For comparing irrational values, use nearby perfect squares when options are sufficiently far apart.

For example, √10 ≈ 3.16 and √11 ≈ 3.32.

Spot Reciprocal Patterns

If you see a^1/n + a^-1/n, substitute t = a^1/n. The expression becomes t + 1/t, which can be squared or cubed easily.

CAT Surds and Indices Questions PDF: How to Use Practice Material Effectively

A Surds and Indices questions PDF is useful only when you solve it strategically. Avoid solving a large number of random questions in one sitting. Divide your preparation into three stages.

Stage 1: Concept Building

Solve direct questions on exponent laws, simplification, and rationalisation. Focus on accuracy rather than speed.

Stage 2: Pattern Recognition

Practise mixed questions involving indices, surds, logarithms, and algebraic identities. Learn to identify the shortest method.

Stage 3: Timed Application

Solve CAT-level questions under a strict time limit. Aim to spend around 90 to 120 seconds on a medium-difficulty question. If the approach is not clear within 30 to 40 seconds, mark it and move ahead.

Surds and Indices Questions for CAT: Practice Plan

Day	Focus Area	Target
Day 1	Laws of indices	25 basic questions
Day 2	Fractional and negative indices	25 questions
Day 3	Simplification of surds	20 questions
Day 4	Rationalisation and conjugates	20 questions
Day 5	Nested surds and identities	15 questions
Day 6	Mixed CAT-level questions	20 timed questions
Day 7	Error analysis and revision	Re-solve incorrect questions

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Common Mistakes to Avoid in CAT Surds and Indices Questions

• Treating √(a + b) as √a + √b
• Forgetting that a⁰ = 1, where a ≠ 0
• Applying exponent rules to different bases incorrectly
• Using decimal approximations too early
• Expanding complex expressions before checking for identities
• Spending too long on a calculation-heavy question in the exam
• Not attempting the CAT mock tests

Conclusion

CAT Surds and Indices Questions can become a reliable scoring area in CAT Quant preparation if you focus on structure rather than calculation. Build fluency with exponent laws, learn to rationalise quickly, convert terms into common bases, and practise recognising standard patterns.

For CAT 2026, aim to solve basic and moderate questions from this topic with near-perfect accuracy. In the exam, prioritise questions where the structure is visible quickly and avoid getting trapped in lengthy algebraic manipulation.