How to Solve: Square Root and Cube Root Problems

How to Solve: Square Root and Cube Root Problems

Complete Guide for SSC / Bank / Railway Exams

? Introduction

"Master square and cube roots, and you’ll unlock 5–10 easy marks in SSC, Bank, or Railway exams—questions that take less than 30 seconds if you know the shortcuts!

? WHAT YOU NEED TO KNOW FIRST

1. Perfect squares and cubes (e.g., 1²=1, 2²=4, 1³=1, 2³=8).
2. Prime factorization (breaking numbers into prime factors like 2×2×3).
3. Basic arithmetic (addition, subtraction, multiplication, division).

? KEY VOCABULARY

✏️ FORMULAS TO KNOW

1. Square Root Formula
2. √(a × b) = √a × √b

3. MEMORISE THIS – Used to simplify roots.

4. Cube Root Formula

5. ∛(a × b) = ∛a × ∛b

6. MEMORISE THIS – Used to simplify cube roots.

7. Approximation (for non-perfect roots)

8. √x ≈ y + (x – y²) / (2y + 1) (where y is the nearest perfect square)
9. Given on exam sheet – Useful for quick estimates.

? STEP-BY-STEP METHOD

How to Find Square Roots

Step 1: Check if the number is a perfect square. - If yes, write the square root directly (e.g., √36 = 6). - If no, proceed to Step 2.

Step 2: Prime factorize the number. - Break it into prime factors (e.g., 72 = 2 × 2 × 2 × 3 × 3).

Step 3: Pair identical factors. - For square roots, group factors in pairs (e.g., 72 = (2×2) × (3×3) × 2).

Step 4: Take one from each pair outside the root. - √72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2.

Step 5: Simplify if possible. - If no pairs remain, leave the root as is (e.g., √5 = √5).

How to Find Cube Roots

Step 1: Check if the number is a perfect cube. - If yes, write the cube root directly (e.g., ∛64 = 4). - If no, proceed to Step 2.

Step 2: Prime factorize the number. - Break it into prime factors (e.g., 54 = 2 × 3 × 3 × 3).

Step 3: Group identical factors in threes. - For cube roots, group factors in triplets (e.g., 54 = 2 × (3×3×3)).

Step 4: Take one from each triplet outside the root. - ∛54 = ∛(2 × 3³) = 3 × ∛2 = 3∛2.

Step 5: Simplify if possible. - If no triplets remain, leave the root as is (e.g., ∛7 = ∛7).

✅ WORKED EXAMPLES

Example 1 – Basic Square Root

Question: Simplify √50. Solution:
1. Prime factorize: 50 = 2 × 5 × 5.
2. Pair identical factors: (5 × 5) × 2.
3. Take one from each pair: 5 × √2.
4. Final answer: 5√2.

What we did and why: - We broke 50 into prime factors, paired the 5s, and took one 5 outside the root.

Example 2 – Medium Cube Root

Question: Simplify ∛108. Solution:
1. Prime factorize: 108 = 2 × 2 × 3 × 3 × 3.
2. Group in threes: (3 × 3 × 3) × (2 × 2).
3. Take one from each triplet: 3 × ∛(2 × 2).
4. Final answer: 3∛4.

What we did and why: - We grouped the 3s in a triplet, took one 3 outside, and left the remaining 2×2 inside.

Example 3 – Exam-Style (Time-Pressure)

Question: If √(x + 5) = 7, find x. Solution:
1. Square both sides: (√(x + 5))² = 7² → x + 5 = 49.
2. Solve for x: x = 49 – 5 = 44.

What we did and why: - We eliminated the square root by squaring both sides, then solved the simple equation.

❌ Common Mistakes

? EXAM TRAPS