Squares And Square Roots — for Class 8

What Are Squares and Square Roots?

Take any number — say, 7. Multiply it by itself: 7 × 7 = 49. That result, 49, is called the square of 7. We write it as 7² = 49.

Now flip the question: which number, when multiplied by itself, gives 49? The answer is 7. So 7 is the square root of 49. We write this as √49 = 7.

This back-and-forth relationship is the entire foundation of this chapter. Squaring and finding square roots are inverse operations — they undo each other, the same way addition and subtraction do.

Why does Class 8 bother with this? Because squares and square roots appear everywhere in higher classes: Pythagoras’ theorem (Class 10), quadratic equations (Class 10), coordinate geometry, and even physics problems involving velocity and kinetic energy. Getting comfortable with them now saves enormous time later.

The good news: this is one of the most scoring topics in Class 8 boards. The methods are systematic, the patterns are recognizable, and with enough practice, you’ll solve most problems mentally.

Key Terms and Definitions

Perfect square — A number that is the square of a whole number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Notice that 15 is not a perfect square — no whole number multiplied by itself gives 15.

Square root (√) — The value which, when squared, gives the original number. √36 = 6 because 6² = 36.

Radical sign (√) — The symbol used to denote square root.

Irrational number — A square root that cannot be expressed as a fraction. √2, √3, √5 are irrational. Perfect squares always give rational (whole number) square roots.

Prime factorization — Breaking a number into its prime factors. This is the most reliable method to find square roots without a calculator, and the method NCERT tests most often.

Memorize perfect squares up to 30² = 900. In board exams, knowing that 529 = 23² instantly saves 2-3 minutes per question.

Properties of Perfect Squares

Before learning methods, these properties help you check your answers and eliminate wrong options in MCQs.

1. Unit Digit Patterns

The unit digit of a perfect square can only be: 0, 1, 4, 5, 6, 9.

A number ending in 2, 3, 7, or 8 is never a perfect square. So 432, 1537, and 2678 are definitely not perfect squares — you know this in one second.

Unit digit of number	Unit digit of its square
1 or 9	1
2 or 8	4
3 or 7	9
4 or 6	6
5	5
0	0

2. Number of Zeros

A perfect square with trailing zeros always has an even number of zeros. So 100 (2 zeros) and 10000 (4 zeros) are perfect squares, but 1000 (3 zeros) is not.

3. Odd and Even

The square of an even number is always even. The square of an odd number is always odd.

4. Sum of Odd Numbers

Every perfect square is the sum of consecutive odd numbers starting from 1:

1 = 1
4 = 1 + 3
9 = 1 + 3 + 5
16 = 1 + 3 + 5 + 7
n² = sum of first n odd numbers

This property is occasionally tested in “fill in the blanks” questions.

Methods for Finding Square Roots

Method 1: Prime Factorization

When to use: For perfect squares, especially in board exams. Always your first choice.

How it works:

Find the prime factorization of the number
Pair up identical prime factors
Take one factor from each pair
Multiply those factors — that’s your square root

Worked example: Find √1764

Step 1: Prime factorize 1764

1764 = 2 \times 882 = 2 \times 2 \times 441 = 2^2 \times 441

441 = 3 \times 147 = 3 \times 3 \times 49 = 3^2 \times 7^2

\therefore 1764 = 2^2 \times 3^2 \times 7^2

Step 2: Take one from each pair

\sqrt{1764} = 2 \times 3 \times 7 = 42

Check: 42² = 1764 ✓

In CBSE boards, prime factorization carries full marks only if you show the factor tree or division method clearly. Don’t just write the answer — show each step.

Method 2: Long Division Method

When to use: For large numbers, or when the number isn’t a perfect square and you need an approximate value.

How it works (for √784 as example):

784 → pair as 7 | 84 (groups of 2 from the right)

Largest square ≤ 7 is 4 (since 2² = 4). Write 2 as divisor and quotient. Remainder = 7 − 4 = 3.

Bring down 84. New dividend = 384.

Double quotient (2) = 4. Now find digit X such that (40 + X) × X ≤ 384. Try X = 8: 48 × 8 = 384.

Quotient = 28. Therefore √784 = 28.

Check: 28² = 784 ✓

This method always works. Practice it for at least 5-6 different numbers before the exam.

Method 3: Repeated Subtraction

When to use: Only for small perfect squares. This is more of a concept-checker than a practical tool.

Subtract successive odd numbers (1, 3, 5, 7…) from n until you reach 0. The count of subtractions = √n.

Example: √25 → 25−1=24, 24−3=21, 21−5=16, 16−7=9, 9−9=0. Count = 5. So √25 = 5. ✓

Checking if a Number is a Perfect Square

To check whether a number is a perfect square using prime factorization:

Find prime factorization
Every prime factor must appear an even number of times

If any prime appears an odd number of times → not a perfect square.

Example: Is 2028 a perfect square?

2028 = 2^2 \times 3 \times 169 = 2^2 \times 3 \times 13^2

The prime 3 appears only once (odd count). So 2028 is not a perfect square.

To make it a perfect square: Multiply by 3 → 2028 × 3 = 6084 = 2² × 3² × 13². Now √6084 = 2 × 3 × 13 = 78.

Solved Examples

Example 1 (CBSE — Easy)

Q: Find the smallest number by which 180 must be multiplied to make it a perfect square. Also find the square root of the result.

Solution:

180 = 2^2 \times 3^2 \times 5

The prime 5 appears once. We need one more 5. Multiply by 5.

180 \times 5 = 900 = 2^2 \times 3^2 \times 5^2

\sqrt{900} = 2 \times 3 \times 5 = 30

Example 2 (CBSE — Easy)

Q: Find the smallest number by which 252 must be divided to make it a perfect square.

Solution:

252 = 2^2 \times 3^2 \times 7

The prime 7 appears once. Divide by 7.

252 \div 7 = 36 = 6^2

\sqrt{36} = 6

Example 3 (CBSE — Medium)

Q: Find √5929 using the long division method.

Solution:

Group: 59 | 29

Largest square ≤ 59: 7² = 49. Quotient = 7, Remainder = 10.
Bring down 29 → 1029. New divisor base = 7 × 2 = 14.
Find X: (140 + X) × X ≤ 1029. Try X = 7: 147 × 7 = 1029.

\sqrt{5929} = 77

Check: 77² = (80−3)² = 6400 − 480 + 9 = 5929 ✓

Example 4 (CBSE — Medium/Hard)

Q: 2025 plants are to be planted in a garden in such a way that the number of rows equals the number of plants in each row. Find the number of rows.

Solution:

We need √2025.

2025 = 5^2 \times 81 = 5^2 \times 3^4

\sqrt{2025} = 5 \times 3^2 = 5 \times 9 = 45

Number of rows = 45, with 45 plants in each row.

Exam-Specific Tips

CBSE Class 8 Boards: This chapter contributes 8-10 marks in a typical paper. Expect: (1) one “find square root by prime factorization” question for 2 marks, (2) one long division problem for 3 marks, (3) one word problem about rows/columns or area for 3 marks. The word problem always involves finding √n of a perfect square.

For word problems: The moment you see “square plot”, “square garden”, “arranged in rows and columns with rows = columns” — it’s a square root problem. Find √(area) or √(total items).

For “smallest number to multiply/divide”: Always do prime factorization first. Unpaired primes tell you what’s needed. Multiply if you need more of an unpaired prime; divide if you want to remove it.

Step marking in boards: If you get the final answer wrong but show correct prime factorization, you’ll still get 1-2 marks. Always write every step.

Estimating square roots: Between which two consecutive integers does √200 lie? 14² = 196, 15² = 225. So √200 is between 14 and 15, closer to 14. CBSE sometimes asks this type of estimation question.

Common Mistakes to Avoid

Mistake 1: Incomplete pairing in prime factorization Students write 36 = 2 × 2 × 3 × 3, then take √36 = 2 + 3 = 5 (wrong!) instead of 2 × 3 = 6. You multiply the factors taken from each pair — never add them.

Mistake 2: Wrong grouping in long division Always group digits in pairs from the right. For √14400: group as 1 | 44 | 00, not 14 | 40 | 0. Getting the first group wrong cascades into a completely wrong answer.

Mistake 3: Assuming every number ending in 4 or 9 is a perfect square 144 is a perfect square (12²). But 124 is not. Don’t reverse the unit digit rule — it tells you what CAN’T be a perfect square, not what IS one.

Mistake 4: Forgetting to check by squaring Always verify: if you find √x = n, check that n² = x. This 5-second step catches calculation errors before the examiner does.

Mistake 5: Misreading “multiply to make perfect square” vs “divide to make perfect square” Read the question twice. If 252 = 2² × 3² × 7, multiply by 7 makes 252 × 7 = 1764 (also perfect square), but dividing by 7 gives 36 (simpler). The question will specify which operation — don’t guess.

Practice Questions

Q1. Find √7056 using prime factorization.

7056 = 2⁴ × 3² × 7² √7056 = 2² × 3 × 7 = 4 × 3 × 7 = 84 Check: 84² = 7056 ✓

Q2. Is 1250 a perfect square? If not, find the smallest number to multiply to make it one.

1250 = 2 × 5⁴ The prime 2 appears once (odd). Multiply by 2. 1250 × 2 = 2500 = 2² × 5⁴ √2500 = 2 × 25 = 50

Q3. Find the smallest number to divide 1620 by so that the quotient is a perfect square. Also find the square root of the quotient.

1620 = 2² × 3⁴ × 5 The prime 5 appears once. Divide by 5. 1620 ÷ 5 = 324 = 2² × 3⁴ √324 = 2 × 9 = 18

Q4. A school hall has 2916 tiles. If the tiles are arranged in a square, how many tiles are there in each row?

We need √2916. 2916 = 4 × 729 = 2² × 3⁶ √2916 = 2 × 3³ = 2 × 27 = 54 Each row has 54 tiles.

Q5. Find √10404 using the long division method.

Group: 1 | 04 | 04

Largest square ≤ 1: 1² = 1. Quotient = 1, Remainder = 0.
Bring down 04 → 004. New divisor base = 1 × 2 = 2. Find X: (20 + X) × X ≤ 4 → X = 0. 200 × 0 = 0.
Bring down 04 → 404. New divisor base = 102 × 2 = 204. Find X: (2040 + X) × X ≤ 404. Wait — re-check grouping.

Correct grouping: 1 | 04 | 04

Step 1: √1 = 1, Quotient: 1, R = 0
Step 2: Bring 04 → 004, double quotient = 2, find X: (20+X)×X ≤ 4, X=0, R=4
Step 3: Bring 04 → 404, double quotient (10) = 20, find X: (200+X)×X ≤ 404, X=2: 202×2=404

√10404 = 102 Check: 102² = 10404 ✓

Q6. Between which two consecutive whole numbers does √200 lie? Which one is closer?

14² = 196, 15² = 225 So √200 lies between 14 and 15. 200 − 196 = 4, 225 − 200 = 25 Since 4 < 25, √200 is closer to 14.

Q7. Without calculating, state whether 4096 is a perfect square. Give two reasons.

Yes, 4096 is likely a perfect square. Reason 1: Unit digit is 6 — squares can end in 6 (4² = 16, 6² = 36, etc.) Reason 2: Number of trailing zeros is 0, which is even (not a conclusive check here, but passes). Verification: 4096 = 2¹², so √4096 = 2⁶ = 64 ✓

Q8. The area of a square field is 5625 m². Find its perimeter.

Side = √5625 = √(25 × 225) = 5 × 15 = 75 m Perimeter = 4 × 75 = 300 m

FAQs

Q: What is the difference between a square and a perfect square?

Technically, squaring any number gives its “square” — 2.5² = 6.25 is the square of 2.5. But a perfect square specifically means the square of a whole number. So 6.25 is not a perfect square, but 25 (= 5²) is. In Class 8, we mostly work with perfect squares.

Q: Can a negative number have a square root?

No negative number has a real square root, because any number squared (positive or negative) gives a positive result. √(−4) is not defined in real numbers. You’ll encounter imaginary numbers (√(−1) = i) in higher classes, but that’s beyond Class 8.

Q: How do I check if my square root answer is correct?

Square it. If n = √x, then n² must equal x exactly. This takes 10 seconds and is always worth doing in an exam.

Q: Why does the long division method work?

The long division method is essentially an algorithmic version of the identity (a + b)² = a² + 2ab + b². Each step refines your estimate of the root by one more digit. You don’t need to understand the derivation for Class 8 — but knowing why helps you remember the steps correctly.

Q: What if a number has an odd number of digits in prime factorization?

That means it’s not a perfect square. For example, 48 = 2⁴ × 3¹. The 3 appears once (odd). No matter what you do, √48 is irrational. The prime factorization method is your most reliable test.

Q: Is there a quick way to square numbers ending in 5?

Yes — a very useful trick. For any number ending in 5: multiply the tens digit by (tens digit + 1), then attach 25. Example: 75² → 7 × 8 = 56, attach 25 → 5625. Works for 15², 25², 35², 45² and so on.

Q: Do square roots appear in Class 8 other than this chapter?

Yes — in the chapter on Pythagorean triplets and in some mensuration problems where you find the side of a square given its area. The square root skill from this chapter directly feeds those problems.

Square of a number: n² = n × n
Square root: If n² = x, then √x = n
Unit digits of perfect squares: Only 0, 1, 4, 5, 6, 9
Sum of first n odd numbers: 1 + 3 + 5 + … + (2n−1) = n²
Perfect squares 1–30: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900
Quick squaring trick (ending in 5): (a5)² = a(a+1) followed by 25