Surds and Indices Questions for IPMAT (2027) 20 Solved Problems + PDF

Looking for Surds and Indices Questions for IPMAT? This page gives you the key rules, common patterns, and a 20-question solved practice set (plus a downloadable PDF) to strengthen Quant for IPMAT 2027.

If you are preparing for the Integrated Programme in Management Aptitude Test (IPMAT), surds and indices are useful fundamentals to revise while building your Quantitative Aptitude skills. Questions can involve simplifying roots, applying exponent laws, rationalising denominators or comparing expressions. 

The exact number and difficulty of questions in IPMAT 2027 cannot be predicted in advance. Focus on understanding the rules and practising a variety of problems rather than relying on a predicted question count.

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What Are Surds in the IPMAT Exam 2027?

A surd is an irrational root that cannot be simplified into a rational number. Common examples include √2, √3 and ∛5. A root may look complicated but become simpler when its radicand contains a perfect-square or perfect-cube factor.

Expression	Surd	Reason
√2	Yes	It is irrational and cannot be simplified further.
√12 = 2√3	Yes	It simplifies, but the result remains irrational.
√25 = 5	No	The root is a rational integer.
∛7	Yes	It is not the cube root of a perfect cube.

In IPMAT preparation, surds commonly appear in simplification, arithmetic with roots, algebraic identities and rationalisation. A useful first step is to factor the number inside the root and identify perfect powers.

What Are Indices in IPMAT 2027?

Indices, also called exponents or powers, show how many times a number is multiplied by itself. In aⁿ, a is the base and n is the exponent.

For example:

2⁴ = 2 × 2 × 2 × 2 = 16

Fractional indices connect powers and roots:

• a¹/² = √a for non-negative real a.
• a¹/³ = ∛a for real a.
• a⁻ⁿ = 1/aⁿ, provided a is not zero.

Understanding these relationships helps you move between radical notation and exponent notation, which is useful when simplifying mixed expressions.

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IPMAT Surds Questions to Practice 2027

Q1. Simplify √48.

48 = 16 × 3, and 16 is a perfect square.

√48 = √16 × √3 = 4√3

(Writing 48 = 4 × 12 also works, but then you get 2√12 and have to simplify again. Always go for the largest square.)

Q2. Simplify 3√5 + 2√5.

Both terms have √5, so they're like surds. Treat √5 the way you'd treat x in 3x + 2x.

3√5 + 2√5 = (3 + 2)√5 = 5√5

Q3. Simplify √75 − √12.

These don't look alike yet. Simplify each first.

√75 = √(25 × 3) = 5√3

√12 = √(4 × 3) = 2√3

So √75 − √12 = 5√3 − 2√3 = 3√3

Q4. Simplify √50 + √8.

√50 = √(25 × 2) = 5√2

√8 = √(4 × 2) = 2√2

Sum = 7√2

Q5. Simplify √98 − √32.

√98 = √(49 × 2) = 7√2

√32 = √(16 × 2) = 4√2

Difference = 3√2

Q6. Simplify √50 + √8 − √18.

You already have the first two from Q4. Only √18 is new.

√18 = √(9 × 2) = 3√2

Total = (5 + 2 − 3)√2 = 4√2

Q7. Simplify √6 × √24.

Multiply under one root before you simplify anything.

√6 × √24 = √144 = 12

If you'd simplified √24 to 2√6 first, you'd get 2 × (√6)² = 2 × 6 = 12. Same answer, one extra step.

Q8. Simplify √3(√12 + √27).

Clean up the bracket first.

√12 = 2√3 and √27 = 3√3, so the bracket is 5√3.

√3 × 5√3 = 5 × (√3)² = 5 × 3 = 15

Q9. Rationalise 1/√7.

Multiply top and bottom by √7. You're multiplying by 1, so the value doesn't change.

(1/√7) × (√7/√7) = √7/7

Q10. Simplify (√5 − √3)(√5 + √3).

This is (a − b)(a + b) = a² − b² with a = √5 and b = √3.

(√5)² − (√3)² = 5 − 3 = 2

Most students know these laws. The mistakes come from mixing them up under time pressure, like multiplying exponents when you should add them.

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IPMAT Indices Questions to Practice 2027

Q1. Simplify 2³ × 2⁵.

Same base, multiplying, so add the powers.

2³ × 2⁵ = 2⁸ = 256

Q2. Simplify 7⁶ ÷ 7⁴.

Same base, dividing, so subtract.

7⁶ ÷ 7⁴ = 7² = 49

Q3. Evaluate (3²)³.

A power raised to a power: multiply.

(3²)³ = 3⁶ = 729

Compare with Q12. 3² × 3³ would be 3⁵, but (3²)³ is 3⁶. This is the single most common slip in this chapter.

Q4. Simplify 5⁻².

A negative index flips the number into the denominator. It doesn't make the answer negative.

5⁻² = 1/5² = 1/25

Q5. Evaluate 16^(1/2).

A power of 1/2 is a square root.

16^(1/2) = √16 = 4

Q6. Evaluate 27^(2/3).

Read the denominator as the root and the numerator as the power. Take the root first, since it keeps the numbers small.

27^(2/3) = (∛27)² = 3² = 9

Q7. Simplify 81^(3/4).

Same method. The fourth root of 81 is 3, because 3 × 3 × 3 × 3 = 81.

81^(3/4) = (⁴√81)³ = 3³ = 27

Q8. Simplify (2⁴ × 2⁻¹) ÷ 2².

Work only with the exponents: 4 + (−1) − 2 = 1.

So the expression is 2¹ = 2

Q9. Express 0.008 as a power of 5.

Turn the decimal into a fraction, then look for powers of 5.

0.008 = 8/1000 = 1/125

125 = 5³, so 1/125 = 5⁻³

If 8/1000 to 1/125 isn't obvious, divide top and bottom by 8: 8 ÷ 8 = 1 and 1000 ÷ 8 = 125.

Important Surds and Indices Rules for IPMAT Exam 2027

Memorise the following rules, but also practise identifying when each rule applies.

Rule	Formula	Example
Product of powers with the same base	aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2² = 2⁵
Quotient of powers with the same base	aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a ≠ 0	3⁵ ÷ 3² = 3³
Power raised to a power	(aᵐ)ⁿ = aᵐⁿ	(2³)² = 2⁶
Zero exponent	a⁰ = 1, a ≠ 0	7⁰ = 1
Negative exponent	a⁻ⁿ = 1/aⁿ, a ≠ 0	2⁻³ = 1/8
Fractional exponent	aᵐ/ⁿ = ⁿ√(aᵐ), where defined	8²/³ = (∛8)² = 4
Product of square roots	√a × √b = √(ab), for a,b ≥ 0	√2 × √3 = √6
Like surds	p√a + q√a = (p + q)√a	3√2 + 2√2 = 5√2
Rationalising a simple denominator	1/√a = √a/a, a > 0	1/√3 = √3/3

Common Question Patterns in Surds and Indices

Surds and indices questions for IPMAT often test your ability to simplify expressions, apply exponent rules and work with square roots. Understanding these common question patterns will help you solve such questions more accurately and efficiently.

1. Simplifying Surds

Break the number inside the square root into factors, including the largest possible perfect square. This helps simplify the surd.

Example: Simplify √72.

Solution: √72 = √(36 × 2) = 6√2

2. Adding and Subtracting Surds

Simplify each surd first. You can then add or subtract terms that have the same radical part.

Example: Simplify √12 + √27.

Solution: √12 + √27 = 2√3 + 3√3 = 5√3

3. Applying the Laws of Indices

Use exponent rules to simplify expressions involving powers. Check whether the bases are the same, and pay attention to negative exponents and brackets.

Example: Simplify 2³ × 2⁴.

Solution: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

4. Rationalising the Denominator

When a denominator contains a surd, multiply the numerator and denominator by the appropriate factor to remove the root from the denominator. For a binomial surd, use its conjugate.

Example: Rationalise 1/(√5 − √2).

Solution:

1/(√5 − √2) × (√5 + √2)/(√5 + √2)

= (√5 + √2)/(5 − 2)

= (√5 + √2)/3

5. Converting Roots into Fractional Powers

Express roots as fractional exponents to simplify expressions involving both roots and powers.

Example: Express ⁴√a³ as a fractional power.

Solution: ⁴√a³ = a³⁄⁴, for (a \geq 0) in the real-number system.

6. Simplifying Expressions Involving Surds and Indices

Apply the BODMAS rule to solve mixed expressions involving brackets, powers, roots and arithmetic operations. Simplify the powers and surds before carrying out the remaining calculations.

Example: Simplify 2² + √16 × 3.

Solution: 4 + 4 × 3 = 4 + 12 = 16

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Common Mistakes to Avoid while Solving Surds and Indices Questions for IPMAT

• Adding unlike surds: √2 + √3 cannot be simplified to √5.
• Ignoring factorisation: Simplify roots before combining them. For example, √12 = 2√3.
• Misusing exponent laws: Exponents are added when multiplying powers with the same base, not when multiplying different bases.
• Forgetting negative powers: a⁻ⁿ = 1/aⁿ for a non-zero base, not −aⁿ.
• Skipping domain conditions: Roots and fractional powers need appropriate real-number conditions.
• Rushing calculations: Keep brackets visible and check signs before selecting an answer.

7-Day Study Plan for Surds and Indices Questions for IPMAT 2027

Day	Focus	Practice Target
Day 1	Perfect squares, cubes and root simplification	15 basic questions
Day 2	Adding, subtracting and multiplying surds	15 questions
Day 3	Positive, zero and negative indices	20 short calculations
Day 4	Fractional powers and root conversion	15 mixed questions
Day 5	Rationalisation and conjugates	10–15 questions
Day 6	Mixed surds-and-indices problems	Complete the 20-question set under timed conditions
Day 7	Error review and revision	Redo every incorrect question without notes

Adjust the targets to your schedule. Consistent, carefully reviewed practice is more valuable than rushing through a large number of questions without understanding your mistakes.

How to Prepare for IPMAT 2027 Quantitative Aptitude?

Preparing for the IPMAT 2027 Quantitative Aptitude section requires a strong understanding of basic concepts, regular practice and effective time management. Follow these tips to improve your preparation:

• Strengthen Your Basics: Start with arithmetic and algebra, then move on to advanced topics and mixed-question practice.
• Practise Questions Regularly: Solve topic-wise questions to improve your accuracy and gradually increase the difficulty level.
• Improve Speed and Accuracy: Once you understand the concepts, practise with a timer to build speed without compromising accuracy.
• Analyse Your Mistakes: Maintain an error log to record incorrect answers, understand the reasons behind them and revise the concepts you find difficult.
• Solve Mock Tests and Previous Year Papers: Practise exam-level questions to understand the question types, assess your preparation and identify areas for improvement.
• Follow the Right Exam Pattern: If you are preparing for IPMAT Indore, IPMAT Rohtak or JIPMAT, check the respective official notifications and tailor your preparation to the exam pattern and syllabus.

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Conclusion

Surds and indices become manageable when you recognise the structure of an expression and apply the correct rule.

Learn the formulas, simplify carefully, practise rationalisation and review every error. Use these 20 solved questions as a starting point for your broader IPMAT 2027 preparation.