Quantitative Aptitude – Square Roots & Cube Roots
Learn to extract roots quickly using estimation, factorization, and shortcuts.
1. Understanding Roots
- Square root: \( \sqrt{n} = x \) means \( x^2 = n \).
- Cube root: \( \sqrt[3]{n} = y \) means \( y^3 = n \).
2. Prime Factorization Method
- Express number as product of primes:
For square root pair factors: \( n = p_1^{2a} p_2^{2b}... \) then \( \sqrt{n} = p_1^a p_2^b... \).
- For cube root group factors in triples: \( n = q_1^{3c} q_2^{3d}... \) then \( \sqrt[3]{n} = q_1^c q_2^d... \).
3. Estimation & Approximation
- Locate nearest perfect square/cube. Example: roots of 50: between \( 7^2=49 \) and \( 8^2=64 \) → ~7.07.
- Use linear interpolation for refinement if needed.
4. Shortcut Tricks
- Memorize squares (1–30) and cubes (1–15).
- Use last-digit patterns to validate roots.
- Apply binomial expansion for approximate roots:
\( \sqrt{a^2 + x} ≈ a + \frac{x}{2a} \).
5. Shortcut for Cube Roots
- Look at last three digits to determine unit digit of root.
- Combine hundreds-digit estimate with last-digit rule to approximate quickly.
6. Exam Strategy & Speed
- First check if the number is a perfect square/cube.
- Use prime factorization for exact values.
- For MCQs, estimate root and eliminate distant choices.
- Avoid long division for roots—estimate and validate options.
7. Common Errors to Avoid
- Misplacing decimal in non-perfect roots—always re-estimate scale.
- Incorrect factor grouping—ensure correct pairing for squares and triplets for cubes.
- Relying on table memory—confirm with estimation.
8. Revision Checklist
- Memorize squares up to 30² and cubes up to 15³.
- Practice prime factor extraction for roots.
- Use estimation for non-perfect roots to check MCQ choices.
Summary: Recognize perfect squares/cubes, use prime factorization for exact roots, apply estimation and shortcuts for fast answers, and verify results against MCQ options to maximize accuracy and speed.