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Surds and Indices

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Introduction

Surds and indices expressions को simplify करने में महत्वपूर्ण होते हैं, खासकर जब roots और exponents शामिल हों। Aptitude exams में ये problems आपकी ability test करते हैं कि आप exponent laws और root simplifications कितनी जल्दी apply कर पाते हैं। इन्हें master करने से आप complex expressions को seconds में solve कर सकते हैं।

Pattern: Surds and Indices

Pattern: Surds and Indices

मुख्य idea: Exponents (indices) के laws apply करें और surds (roots) को step-by-step simplify करें।

  • am × an = am+n
  • am ÷ an = am-n
  • (am)n = am×n
  • √a = a1/2, ³√a = a1/3, आदि।

Step-by-Step Example

Question

Simplify: (√8 × √2) ÷ √4

Options:

  • A) 1
  • B) 2
  • C) 4
  • D) 8

Solution

  1. Step 1: Roots को exponent form में बदलें

    Rewrite: √8 = 81/2, √2 = 21/2, √4 = 41/2.

  2. Step 2: Numerator को multiply करें

    √8 × √2 = √(8×2) = √16 = 4.

  3. Step 3: Denominator evaluate करें

    √4 = 2.

  4. Step 4: Final simplification

    4 ÷ 2 = 2.

  5. Final Answer:

    2 → Option B.

  6. Quick Check:

    (√8 × √2) = 2√2 × √2 = 4; 4 ÷ √4 = 4 ÷ 2 = 2.

Quick Variations

1. Roots को fractional exponents में बदलकर power rules apply करें।

2. Cube roots और higher roots को indices से simplify करें।

3. Surds को compare करने के लिए decimals में बदलें या rationalize करें।

4. High powers को reduce करें, जैसे (a1/2)4 = a².

Trick to Always Use

  • Step 1 → Roots को exponents में convert करें (जैसे √a = a1/2).
  • Step 2 → Exponent rules apply करें (add, subtract, multiply exponents).
  • Step 3 → Simplify करके lowest integer या fraction form में लाएँ।

Summary

Surds और indices में:

  • Roots को fractional powers में लिखा जा सकता है।
  • Expressions को simplify करने के लिए exponent laws apply करें।
  • Multiplication/division को एक ही root के अंदर combine कर सकते हैं।
  • Quick answers के लिए हमेशा simplest form में reduce करें।

Practice

(1/5)
1. Simplify: √50
easy
A. 5√2
B. 10√2
C. 25√2
D. 2√5

Solution

  1. Step 1: Factorize inside the root

    50 = 25 × 2.

  2. Step 2: Extract the perfect square

    √50 = √(25×2) = √25 × √2 = 5√2.

  3. Final Answer:

    5√2 → Option A.

  4. Quick Check:

    5√2 ≈ 7.07 and √50 ≈ 7.071 → matches.
Hint: Break the number into (perfect square × other factor), then simplify.
Common Mistakes: Leaving √50 unsimplified or incorrectly splitting factors.
2. Simplify: √72
easy
A. 6√2
B. 8√2
C. 12√2
D. 3√6

Solution

  1. Step 1: Factor inside the root

    72 = 36 × 2.

  2. Step 2: Extract square root

    √72 = √36 × √2 = 6√2.

  3. Final Answer:

    6√2 → Option A.

  4. Quick Check:

    6×1.414 ≈ 8.484 and √72 ≈ 8.485 → matches.
Hint: Always pull out perfect squares like 36, 25, 9, etc.
Common Mistakes: Trying to simplify without factorizing into a perfect square.
3. Simplify: 2³ × 2⁴
easy
A. 64
B. 128
C. 32
D. 16

Solution

  1. Step 1: Apply exponent rule

    a^m × a^n = a^(m+n).

  2. Step 2: Add exponents

    2³ × 2⁴ = 2^(3+4) = 2⁷ = 128.

  3. Final Answer:

    128 → Option B.

  4. Quick Check:

    8 × 16 = 128 → correct.
Hint: When multiplying same bases, add exponents.
Common Mistakes: Multiplying exponents instead of adding.
4. Simplify: √18 ÷ √2
medium
A. 2
B. √2
C. 3
D. 3/2

Solution

  1. Step 1: Combine surds

    √18 ÷ √2 = √(18/2) = √9.

  2. Step 2: Evaluate

    √9 = 3.

  3. Final Answer:

    3 → Option C.

  4. Quick Check:

    4.243 ÷ 1.414 ≈ 3 → correct.
Hint: Divide surds by combining inside one root: √a / √b = √(a/b).
Common Mistakes: Simplifying each root separately instead of combining.
5. Simplify: (√5)⁴
medium
A. 5
B. 10
C. 20
D. 25

Solution

  1. Step 1: Convert to exponent form

    (√5)⁴ = (5^(1/2))⁴.

  2. Step 2: Apply exponent rule

    (a^m)^n = a^(m×n) → 5^(1/2 × 4) = 5².

  3. Step 3: Evaluate

    5² = 25.

  4. Final Answer:

    25 → Option D.

  5. Quick Check:

    (√5)² = 5 → (√5)⁴ = 5² = 25.
Hint: Write surds as fractional powers before applying exponent rules.
Common Mistakes: Mistaking (√5)⁴ for √(5⁴).