Solved Examples

Surds & Indices

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Surds & Indices

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Solved Examples

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Solved Examples – Surds & Indices

Solved examples help students understand the practical application of surds, radicals, exponents, and simplification techniques in competitive examinations. These examples are designed from basic to advanced level and cover important questions frequently asked in SSC, Banking, Railway, Insurance, Defence, CAT, CDS, NDA, and various aptitude examinations.

Topics Covered in Solved Examples

  • Laws of Indices
  • Surd Simplification
  • Fractional Powers
  • Negative Indices
  • Rationalization of Denominators
  • Product and Division of Surds
  • Radical Transformations
  • Advanced Simplification Problems

Example 1: Product Law of Indices

Question: Simplify:

23 × 25

Solution:

Using:

am × an = am+n

23 × 25

= 28

= 256

Answer: 256

Example 2: Division Law of Indices

Question: Simplify:

57 / 53

Solution:

Using:

am / an = am−n

57 / 53

= 54

= 625

Answer: 625

Example 3: Power of a Power

Question: Simplify:

(32)4

Solution:

Using:

(am)n = amn

(32)4

= 38

= 6561

Answer: 6561

Example 4: Zero Index Problem

Question: Simplify:

9990

Solution:

Any non-zero number raised to power zero equals 1.

Therefore:

9990 = 1

Answer: 1

Example 5: Negative Index Problem

Question: Simplify:

2−3

Solution:

Using:

a−n = 1/an

2−3

= 1/23

= 1/8

Answer: 1/8

Example 6: Fractional Power Problem

Question: Simplify:

161/2

Solution:

161/2 = √16

= 4

Answer: 4

Example 7: Simplification of Surds

Question: Simplify:

√72

Solution:

√72 = √(36 × 2)

= √36 × √2

= 6√2

Answer: 6√2

Example 8: Product of Surds

Question: Simplify:

√5 × √20

Solution:

√5 × √20

= √100

= 10

Answer: 10

Example 9: Division of Surds

Question: Simplify:

√50 / √2

Solution:

√50 / √2

= √(50/2)

= √25

= 5

Answer: 5

Example 10: Radical Form Conversion

Question: Express 82/3 in radical form and simplify.

Solution:

82/3 = ∛(82)

= ∛64

= 4

Answer: 4

Example 11: Rationalizing the Denominator

Question: Rationalize:

1/(√3 + 1)

Solution:

Multiply numerator and denominator by:

(√3 − 1)

= (√3 − 1) / [(√3 + 1)(√3 − 1)]

= (√3 − 1)/(3 − 1)

= (√3 − 1)/2

Answer: (√3 − 1)/2

Example 12: Conjugate Surd Problem

Question: Find the conjugate of:

5 + √7

Solution:

Conjugate changes sign between terms.

Therefore:

Conjugate = 5 − √7

Answer: 5 − √7

Example 13: Cube Root Simplification

Question: Simplify:

∛54

Solution:

∛54 = ∛(27 × 2)

= ∛27 × ∛2

= 3∛2

Answer: 3∛2

Example 14: Decimal Root Problem

Question: Simplify:

√0.25

Solution:

√0.25 = √(25/100)

= 5/10

= 1/2

Answer: 1/2

Example 15: Advanced Simplification Problem

Question: Simplify:

(√5 + √3)(√5 − √3)

Solution:

Using identity:

(a + b)(a − b) = a2 − b2

= (√5)2 − (√3)2

= 5 − 3

= 2

Answer: 2

Important Exam Tips

  • Memorize all laws of indices thoroughly.
  • Learn perfect squares and cubes.
  • Practice radical simplification regularly.
  • Use conjugates correctly for rationalization.
  • Convert fractional powers into radicals quickly.
  • Improve simplification speed.
  • Practice previous year aptitude questions.

Common Mistakes to Avoid

  • Ignoring negative powers.
  • Using incorrect surd simplification.
  • Forgetting conjugates during rationalization.
  • Calculation mistakes in powers.
  • Confusing radicals and rational numbers.

Practicing solved examples regularly improves conceptual clarity, calculation speed, and logical analysis in solving Surds & Indices aptitude questions in competitive examinations.

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