Square Roots — Perfect & Non-Perfect Numbers

Last Digit of Squares

The last digit of a number's square depends only on its last digit:

Last digit	Square ends with
0	0
1	1
2	4
3	9
4	6
5	5
6	6
7	9
8	4
9	1

Key pairs: 2↔8 (both end in 4), 3↔7 (both end in 9), 4↔6 (both end in 6), 1↔9 (both end in 1), 5→5, 0→0

---

Finding Square Roots of Perfect Squares (2-Step Method)

Step 1: Identify the range

Find the nearest multiples of 5 or 10 whose squares bracket the number.

Step 2: Use last digit to pick the exact answer

From the last digit of the number, determine which digit the root ends with.

---

Example 1: √2116

• Range: 45² = 2025, 50² = 2500 → root is between 45 and 50
• Last digit of 2116 is 6 → root ends in 4 or 6
• Between 45-50: only 46 ends in 6
• √2116 = 46 ✓

Example 2: √5329

• 70² = 4900, 75² = 5625 → root is between 70 and 75
• Last digit is 9 → root ends in 3 or 7
• 73² ends in 9 ✓ → √5329 = 73

Example 3: √4489

• 65² = 4225, 70² = 4900 → between 65 and 70
• Last digit 9 → ends in 3 or 7
• 67² ends in 9 ✓ → √4489 = 67

Example 4: √7396

• 85² = 7225, 90² = 8100 → between 85 and 90
• Last digit 6 → ends in 4 or 6
• 86² → √7396 = 86

Example 5: √8464

• 90² = 8100, 95² = 9025 → between 90 and 95
• Last digit 4 → ends in 2 or 8
• 92² → √8464 = 92

Example 6: √3481

• 55² = 3025, 60² = 3600 → between 55 and 60
• Last digit 1 → ends in 1 or 9
• 59² → √3481 = 59

Example 7: √2916

• 50² = 2500, 55² = 3025 → between 50 and 55
• Last digit 6 → ends in 4 or 6
• 54² → √2916 = 54

Example 8: √6084

• 75² = 5625, 80² = 6400 → between 75 and 80
• Last digit 4 → ends in 2 or 8
• 78² → √6084 = 78

---

Squaring Numbers Ending in 5 (Instant Shortcut)

For any number ending in 5: n5² = n(n+1) | 25

Write n × (n+1) followed by 25.

Number	n × (n+1)	Result
15²	1 × 2 = 2	225
25²	2 × 3 = 6	625
35²	3 × 4 = 12	1225
45²	4 × 5 = 20	2025
55²	5 × 6 = 30	3025
65²	6 × 7 = 42	4225
75²	7 × 8 = 56	5625
85²	8 × 9 = 72	7225
95²	9 × 10 = 90	9025

This shortcut makes finding ranges for square roots much faster — you instantly know the "bookend" perfect squares.

---

Square Roots of Non-Perfect Numbers (Approximation)

The Formula:

√(a ± b) ≈ √a ± b/(2√a)

Where a is the nearest perfect square and b is the difference.

Example 1: √210

• Nearest perfect square: 196 (= 14²). Extra: 210 − 196 = 14
• √210 = √196 + 14/(2×14) = 14 + 14/28 = 14 + 0.5 = 14.5

Example 2: √264

• Nearest: 256 (= 16²). Extra: 264 − 256 = 8
• √264 = 16 + 8/(2×16) = 16 + 8/32 = 16 + 0.25 = 16.25

Example 3: √568

• Nearest: 576 (= 24²). Deficit: 576 − 568 = 8
• √568 = 24 − 8/(2×24) = 24 − 8/48 = 24 − 0.166 = 23.83

Example 4: √717

• Nearest: 729 (= 27²). Deficit: 729 − 717 = 12
• √717 = 27 − 12/(2×27) = 27 − 12/54 = 27 − 0.222 = 26.78

When to use: This approximation is accurate enough for exam-level simplification and DI questions where exact values aren't needed.

---

Practice Problems

#	Problem	Answer
1	√3481	59
2	√6084	78
3	√8464	92
4	55² (instant)	3025
5	85² (instant)	7225
6	√210 (approx)	≈ 14.5
7	√568 (approx)	≈ 23.83
8	√7396	86