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Introduction
Many aptitude questions hide simple squares, cubes or roots inside longer expressions. Recognising perfect squares, perfect cubes and when to take square/cube roots lets you simplify quickly and avoid lengthy calculation.
Pattern: Square & Cube Based Simplification
Key idea: Identify perfect squares/cubes or root expressions, evaluate them first (or convert to powers), then perform remaining arithmetic step by step.
Step-by-Step Example
Question
Simplify: √144 + 5³ - 2²
Options:
- A) 120
- B) 128
- C) 133
- D) 140
Solution
-
Step 1: Evaluate perfect powers
√144 = 12 (since 12×12 = 144), 5³ = 125 (5×5×5), 2² = 4. -
Step 2: Substitute values
Expression becomes: 12 + 125 - 4. -
Step 3: Perform arithmetic
12 + 125 = 137; 137 - 4 = 133. -
Final Answer:
133 → Option C. -
Quick Check:
√144 ≈ 12, 125, 4 → 12 + 125 - 4 = 133 ✅
Quick Variations
1. Mixed: Evaluate roots and powers first in expressions like √81 + 2³ × 3.
2. Fractional powers: a^(1/2) = √a, a^(1/3) = ∛a - evaluate when perfect.
3. Nested powers: (2²)³ = 2^(2×3) = 2⁶ - multiply exponents when base repeats.
4. Use factorization for simplification: √(36×5) = 6√5 when exact root extraction is partial.
Trick to Always Use
- Step 1 → Scan for perfect squares (1,4,9,16,25,36,49,64,81,100,121,144,...) and perfect cubes (1,8,27,64,125,...).
- Step 2 → Evaluate those powers/roots immediately to reduce expression size before arithmetic.
- Step 3 → When exponents share the same base, use rules: a^m × a^n = a^(m+n), (a^m)^n = a^(m×n).
Summary
For square & cube based simplification:
- Always spot and evaluate perfect squares/cubes and exact roots first.
- Use exponent rules to simplify repeated bases before numeric calculation.
- Convert roots into fractional exponents when helpful (√ = ^1/2, ∛ = ^1/3).
- Quick-check answers by approximating roots and powers to ensure arithmetic is correct.