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Square Roots – Methods to Find Square Roots

🎓 Class 8 Mathematics CBSE Theory Ch 1 — Squares, Square Roots, Cubes and Cube Roots ⏱ ~35 min
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This MCQ module is based on: Square Roots – Methods to Find Square Roots

This mathematics assessment will be based on: Square Roots – Methods to Find Square Roots
Targeting Class 8 level in Number Theory, with Basic difficulty.

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Square Root↗ — What and Why

A square has area 49 sq. cm. What is the length of its side? We need the number whose square is 49.

7 side = 7 cm n = 7 → squaring (n²) 49 area = 49 sq cm n² = 49 → square root (√) 7 side = 7 cm √49 = 7
Definition — Square Root
If \(y = x^2\), then \(x\) is called the square root of y, written as \(\sqrt{y} = x\).
Every positive perfect square has two integer square roots — one positive and one negative:
\(\sqrt{64} = \pm 8\) because \(8^2 = 64\) and \((-8)^2 = 64\).
In this chapter, we consider only the positive square root.

Three Methods to Find Square Roots

Method 1
Sequential Listing

List squares sequentially (20²=400, 21²=441, 22²=484…) until you find the number. Works for small numbers but becomes slow for large values.

Method 2
Successive Odd-Number Subtraction

Subtract consecutive odd numbers starting from 1. The count of subtractions needed to reach 0 is the square root. Only works for perfect squares.

Method 3
Prime Factorisation

Find prime factors. If they can be split into two identical groups, the product of one group is the square root. Most efficient for larger numbers.

Method 2 — Successive Subtraction: Finding √81

Subtract consecutive odd numbers from 81 until reaching 0 81 80 77 72 65 56 45 32 17 −1 −3 −5 −7 −9 −11 −13 −15 −17 81 80 77 72 65 56 45 32 17 0 Reached 0 after 9 subtractions → √81 = 9

Method 3 — Prime Factorisation

A perfect square's prime factors can always be split into two identical groups. The product of one group is the square root.

📝 Is 324 a perfect square? Find √324.
324 = 2 × 162 = 2 × 2 × 81 = 2 × 2 × 3 × 27 = 2 × 2 × 3 × 3 × 9 = 2 × 2 × 3 × 3 × 3 × 3

Group into pairs:

\(324 = \underbrace{(2 \times 2)}_{\text{pair}} \times \underbrace{(3 \times 3)}_{\text{pair}} \times \underbrace{(3 \times 3)}_{\text{pair}}\)

Or: \(324 = (2 \times 3 \times 3) \times (2 \times 3 \times 3) = 18 \times 18\)

∴ √324 = 18 ✓

324 = 2 × 2 × 3 × 3 × 3 × 3 — split into two equal groups Group 1 2 × 3 × 3 = 18 × Group 2 2 × 3 × 3 = 18 ∴ √324 = 18
📝 Is 156 a perfect square?
Prime factorisation: 156 = 2 × 2 × 3 × 13
Pairs: (2×2) paired ✓, but 3 and 13 have no partners ✗
156 is NOT a perfect square (factors cannot be split into two equal groups).

Estimating Square Roots of Non-Squares

When a number is not a perfect square, we can estimate its square root by finding which two consecutive perfect squares it lies between.

📝 Find √1936 (step-by-step estimation)
Step 1: 1600 = 40² and 2500 = 50², so 40 < √1936 < 50
Step 2: Last digit of 1936 is 6 → square root must end in 4 or 6 → candidates: 44 or 46
Step 3: Calculate 45²: (40+5)² = 1600+400+25 = 2025 > 1936 → so √1936 < 45
Step 4: Range narrows to 40 < √1936 < 45 → only candidate is 44
Verify: 44² = 44×44 = 1936 ✓ → √1936 = 44
Estimating √250: locate between consecutive squares 10²=100 15²=225 250 16²=256 20²=400 √250 ≈ 15.8 15 < √250 < 16 (closer to 16)
📝 Akhil's Cloth (Area = 125 cm²)
125 is not a perfect square. Find nearest squares: 11² = 121 and 12² = 144.
So 11 < √125 < 12, meaning the cloth side is between 11 and 12 cm.
Largest square handkerchief with integer side: 11 cm (since 11² = 121 ≤ 125 < 144 = 12²).
🔧 Square Root Analyser

Enter any number to check if it's a perfect square and find/estimate its square root.


Figure it Out — Exercises (Pages 10–11)

Q1. Which of the following numbers are NOT perfect squares?
(i) 2032   (ii) 2048   (iii) 1027   (iv) 1089
Check units digit: perfect squares end in 0,1,4,5,6,9.
(i) 2032 — ends in 2 → NOT a perfect square ✗
(ii) 2048 — ends in 8 → NOT a perfect square ✗
(iii) 1027 — ends in 7 → NOT a perfect square ✗
(iv) 1089 — ends in 9 (possible). Check: 33² = 1089. IS a perfect square ✓
Answer: (i), (ii), and (iii) are not perfect squares.
Q2. Which one among 64², 108², 292², 36² has last digit 4?
The last digit of n² depends on last digit of n:
64 → ends in 4 → 4²=16 → square ends in 6
108 → ends in 8 → 8²=64 → square ends in 4 ✓
292 → ends in 2 → 2²=4 → square ends in 4 ✓
36 → ends in 6 → 6²=36 → square ends in 6
Answer: 108² and 292² both end in 4. (If only one answer expected: 108²)
Q3. Given 125² = 15625, what is the value of 126²?
(i) 15625+126   (ii) 15625+262   (iii) 15625+253   (iv) 15625+251   (v) 15625+512
The 126th odd number = 2×126 − 1 = 251.
126² = 125² + 251 = 15625 + 251 = 15876.
Answer: (iv) 15625 + 251
Q4. Find the length of the side of a square whose area is 441 m².
We need √441. Prime factorisation: 441 = 3 × 3 × 7 × 7 = (3×7)² = 21².
Answer: Side = √441 = 21 m
Q5. Find the smallest square number divisible by each of 4, 9, and 10.
LCM(4, 9, 10): 4=2², 9=3², 10=2×5.
LCM = 2² × 3² × 5 = 180.
180 = 2² × 3² × 5. The factor 5 has no pair, so multiply by 5:
180 × 5 = 900 = 2² × 3² × 5² = 30².
Answer: 900
Q6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.
9408 = 2 × 4704 = 2 × 2 × 2352 = 2² × 2352 = 2² × 2 × 1176 = 2³ × 1176
= 2³ × 4 × 294 = 2⁵ × 294 = 2⁵ × 2 × 147 = 2⁶ × 147 = 2⁶ × 3 × 49 = 2⁶ × 3 × 7²
= (2³)² × 7² × 3 = 64 × 49 × 3.
The unpaired factor is 3. Multiply by 3: 9408 × 3 = 28224.
28224 = 2⁶ × 3² × 7² = (2³ × 3 × 7)² = (8×3×7)² = 168².
Answer: Multiply by 3; product = 28224; √28224 = 168.
Q7. How many numbers lie between the squares of the following?
(i) 16 and 17    (ii) 99 and 100
Between n² and (n+1)² lie exactly 2n numbers.
(i) Between 16² = 256 and 17² = 289: 2×16 = 32 numbers (257 to 288).
(ii) Between 99² = 9801 and 100² = 10000: 2×99 = 198 numbers.
Q8. Fill in the pattern:
1² + 2² + 2² = 3²
2² + 3² + 6² = 7²
3² + 4² + 12² = 13²
4² + 5² + 20² = (__)²
9² + 10² + (__)² = (__)²
Pattern: \(n^2 + (n+1)^2 + [n(n+1)]^2 = [n(n+1)+1]^2\)

For n=4: \(4^2 + 5^2 + (4\times5)^2 = (4\times5+1)^2\)
→ 16 + 25 + 400 = 441 = 21²

For n=9: \(9^2 + 10^2 + (9\times10)^2 = (90+1)^2\)
→ 81 + 100 + 8100 = 8281 = 91²
Missing terms: \((9\times10)^2 = \mathbf{90^2}\) and result = 91²
Q9. How many tiny squares are there in the picture (a 12×12 grid with a 4×4 hole)? Write the prime factorisation of that number.
Based on the described figure (large square minus smaller square): 12² − 4² = 144 − 16 = 128.
Prime factorisation: 128 = 2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2.
Answer: 128 tiny squares; 128 = 2⁷
📝

Competency-Based Questions

A farmer has two square fields. Field A has area 1156 m² and Field B has area 2800 m². He wants to fence both fields and also build a combined square enclosure from leftover materials. Rani, his daughter, has been learning about square roots. She uses prime factorisation to solve the problems without a calculator.
Q1. Is 1156 a perfect square? Using the units digit rule, what can you say about it? L1 Remember
  • (A) No — ends in 6, so not a perfect square
  • (B) Yes — ends in 6, which is a possible units digit; 34² = 1156
  • (C) Cannot determine from units digit alone
  • (D) No — ends in an odd digit
Answer: (B)
1156 ends in 6 — possible for a square. Verify: 34² = 1156. ✓ So the side of Field A is 34 m.
Q2. Is 2800 a perfect square? Use prime factorisation to determine, and if not, what is the smallest multiplier to make it a perfect square? L3 Apply
2800 = 2 × 1400 = 2² × 700 = 2² × 4 × 175 = 2⁴ × 175 = 2⁴ × 5² × 7.
The factor 7 has no pair. So 2800 is not a perfect square.
Multiply by 7: 2800 × 7 = 19600 = 2⁴ × 5² × 7² = (2² × 5 × 7)² = 140².
Side of combined enclosure = √19600 = 140 m.
Q3. Rani estimates √2800 without a calculator. Between which two consecutive integers does it lie? L2 Understand
52² = 2704 and 53² = 2809.
Since 2704 < 2800 < 2809, we get 52 < √2800 < 53.
Since 2800 is much closer to 2809, √2800 ≈ 52.9.
Q4. The total fencing needed for Field A (side 34 m) is how much? Using the difference of squares, find the fencing for a field with area 1225 m² — without computing √1225 from scratch. L4 Analyse
Field A fencing = 4 × 34 = 136 m.
For area 1225: note 35² = 1225 (since 34²=1156 and the 35th odd number = 2×35−1=69; 1156+69=1225 ✓).
So √1225 = 35; fencing = 4 × 35 = 140 m.
Q5 (HOT). Rani says: "If I find a number with exactly 3 factors, it must be a perfect square." Is she correct? Justify with examples and counter-examples. L5 Evaluate
Yes, Rani is correct.
A number with exactly 3 factors must be of the form p² (square of a prime), because:
— Its factors are 1, p, and p²
— Examples: 4 = 2² (factors: 1,2,4), 9 = 3² (factors: 1,3,9), 25 = 5² (factors: 1,5,25), 49 = 7² (factors: 1,7,49).
A number with 3 factors is always a perfect square of a prime, so all numbers with exactly 3 factors are perfect squares. ✓
🔍 Assertion–Reason Questions

A: Both true; R explains A.   B: Both true; R does NOT explain A.   C: A true, R false.   D: A false, R true.

Assertion: √324 = 18.
Reason: 324 = 2² × 3⁴, and the prime factors can be split into two equal groups each giving 2 × 3² = 18.
Answer: A — Both true; R correctly explains A via prime factorisation method.
Assertion: The square root of a perfect square is found by the successive odd-number subtraction method in exactly √n steps.
Reason: Every perfect square equals the sum of consecutive odd numbers starting from 1.
Answer: A — Both are true; R is the exact reason A works — since n² = 1+3+5+…+(2n−1), subtracting those odd numbers one by one takes exactly n steps to reach 0.
Assertion: 576 is a perfect square with square root 24.
Reason: 576 ends in 6, which is a possible units digit for a perfect square, so it must be a perfect square.
Answer: C — The Assertion is true (24² = 576 ✓). The Reason is false — ending in 6 is necessary but not sufficient; for example, 26 ends in 6 but is not a perfect square.

Frequently Asked Questions — Squares, Square Roots, Cubes and Cube Roots

What is Square Roots - Methods to Find Square Roots in NCERT Class 8 Mathematics?

Square Roots - Methods to Find Square Roots is a key concept covered in NCERT Class 8 Mathematics, Chapter 1: Squares, Square Roots, Cubes and Cube Roots. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Square Roots - Methods to Find Square Roots step by step?

To solve problems on Square Roots - Methods to Find Square Roots, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 8 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 1: Squares, Square Roots, Cubes and Cube Roots?

The essential formulas of Chapter 1 (Squares, Square Roots, Cubes and Cube Roots) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Square Roots - Methods to Find Square Roots important for the Class 8 board exam?

Square Roots - Methods to Find Square Roots is part of the NCERT Class 8 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Square Roots - Methods to Find Square Roots?

Common mistakes in Square Roots - Methods to Find Square Roots include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Square Roots - Methods to Find Square Roots?

End-of-chapter NCERT exercises for Square Roots - Methods to Find Square Roots cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 1, and solve at least one previous-year board paper to consolidate your understanding.

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