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Surds and Indices

Introduction

Surds and indices are important for simplifying expressions involving roots and exponents. In aptitude exams, these problems test your ability to quickly apply exponent laws and root simplifications. Mastering them helps in handling complex expressions in seconds.

Pattern: Surds and Indices

Pattern

Key idea: Apply the laws of exponents (indices) and simplify surds (roots) step by step.

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

Step-by-Step Example

Question

Simplify: (√8 × √2) ÷ √4

Options:

  • A) 1
  • B) 2
  • C) 4
  • D) 8

Solution

  1. Step 1: Convert roots into exponent form

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

  2. Step 2: Multiply numerator

    √8 × √2 = √(8×2) = √16 = 4.

  3. Step 3: Evaluate denominator

    √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. Convert roots into fractional exponents and apply power rules.

2. Simplify cube roots and higher roots using indices.

3. Compare surds by converting them into decimals or rationalizing.

4. Reduce higher powers like (a1/2)4 = a².

Trick to Always Use

  • Step 1 → Convert roots into exponents (like √a = a1/2).
  • Step 2 → Apply exponent rules (add, subtract, multiply exponents).
  • Step 3 → Simplify to lowest integer or fraction form.

Summary

Summary

In surds and indices:

  • Roots can be expressed as fractional powers.
  • Apply exponent laws to simplify expressions.
  • Combine multiplication/division under a single root when possible.
  • Always reduce to the simplest form for quick answers.

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).

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