Class 8 Mathematics — Squares and Square Roots | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: If you have 25 square tiles, you can arrange them in a perfect 5 × 5 square floor — because 25 is a perfect square! Architects use square roots to find the side length of a square room when they know its area. The ancient Egyptians used a 3-4-5 right triangle (a Pythagorean triplet) to make perfect right angles for their pyramids. Squares and square roots are everywhere in design, science, and construction.

🔢 Topic 1: What Is a Perfect Square?

A perfect square = a number times itself.
Example: 3 × 3 = 9. So 9 is a perfect square.
We write 3² = 9.
The list starts: 1, 4, 9, 16, 25, 36 ...
1² = 1, 2² = 4, 3² = 9, 4² = 16.
5² = 25, 6² = 36, 7² = 49.
A perfect square is always whole.

🔢 Topic 2: Properties of Perfect Squares

A perfect square ends in 0, 1, 4, 5, 6, or 9.
It never ends in 2, 3, 7, or 8.
If a number ends in 2, it is not a perfect square.
A perfect square has an even number of zeros at the end.
Example: 400 ends in two zeros. 400 = 20². Yes!
Example: 8000 ends in three zeros. Not a perfect square.

🔢 Topic 3: Patterns in Squares

1 = 1 = 1² (first odd number).
1 + 3 = 4 = 2² (sum of first 2 odd numbers).
1 + 3 + 5 = 9 = 3² (sum of first 3 odd numbers).
1 + 3 + 5 + 7 = 16 = 4².
Rule: Sum of first n odd numbers = n².
This is a beautiful pattern!
Also: (n+1)² − n² = 2n + 1 (always odd).

Sum of first n odd numbers = n²

🔢 Topic 4: Pythagorean Triplets

Three numbers a, b, c form a Pythagorean triplet.
The rule is: a² + b² = c².
Example: 3² + 4² = 9 + 16 = 25 = 5².
So 3, 4, 5 is a Pythagorean triplet.
Other triplets: 5, 12, 13 and 8, 15, 17.
Formula: For any m > 1:
2m, m² − 1, m² + 1 is a triplet.

a² + b² = c² (Pythagorean Triplet)

🔢 Topic 5: What Is a Square Root?

Square root is the opposite of squaring.
If 5² = 25, then √25 = 5.
If 7² = 49, then √49 = 7.
The symbol √ means square root.
√1 = 1, √4 = 2, √9 = 3, √16 = 4.
We find the number that was squared.

If n² = m, then √m = n

🔢 Topic 6: Square Root by Repeated Subtraction

We subtract odd numbers one by one.
Start from 1, then 3, then 5, then 7 ...
Count how many times until we reach zero.
Example: √16 by repeated subtraction:
16 − 1 = 15, 15 − 3 = 12, 12 − 5 = 7, 7 − 7 = 0.
We subtracted 4 times. So √16 = 4.
This works for small perfect squares.

🔢 Topic 7: Square Root by Prime Factorisation

Write the number as a product of primes.
Make pairs of same prime factors.
Take one from each pair.
Multiply them = square root.
Example: 144 = 2 × 2 × 2 × 2 × 3 × 3.
Pairs: (2,2), (2,2), (3,3).
Take one each: 2 × 2 × 3 = 12. So √144 = 12.

🔢 Topic 8: Square Root by Long Division

This method works for big numbers.
Step 1: Group digits in pairs from the right.
Step 2: Find largest number whose square ≤ first group.
Step 3: Subtract and bring down next pair.
Step 4: Double the quotient. Find next digit.
Step 5: Repeat until all pairs are done.
We will see a full example in Section 3.

🔢 Topic 9: Square Root of Decimals

For decimals, we use long division.
Group whole part in pairs from right.
Group decimal part in pairs from left.
Example: √6.25 → groups are 6 and 25.
Then do long division as normal.
Place the decimal point in the answer.
√6.25 = 2.5. Check: 2.5 × 2.5 = 6.25.

🔢 Topic 10: Estimating Square Roots

Not every number is a perfect square.
We can estimate the square root.
Find which two perfect squares it is between.
Example: √50 is between √49 and √64.
So √50 is between 7 and 8.
50 is closer to 49, so √50 ≈ 7.07.
This helps us guess before we calculate.

🎨 Section 2 — Easy Illustrations

Picture 1: Perfect Squares as Dot Arrays

Dots arranged in squares: 1×1, 2×2, 3×3, 4×4. That is why they are called "perfect squares".

Picture 2: Sum of Odd Numbers Pattern

Add odd numbers one by one. You always get a perfect square!

Picture 3: Pythagorean Triplet (3, 4, 5)

In a right triangle, the square of the longest side equals the sum of the squares of the other two sides.

Picture 4: Long Division for √529

Click "Play" to see each step of the long division method. It works for any number!

Picture 5: Square Roots on a Number Line

√2 and √3 sit between 1 and 2. √5 sits between 2 and 3. They are not exact whole numbers.

✏️ Section 3 — Worked Examples

Example 1: Find √784 by Prime Factorisation

Question: Find the square root of 784.

1 Divide 784 by 2: 784 ÷ 2 = 392.

2 Divide 392 by 2: 392 ÷ 2 = 196.

3 Divide 196 by 2: 196 ÷ 2 = 98.

4 Divide 98 by 2: 98 ÷ 2 = 49.

5 Divide 49 by 7: 49 ÷ 7 = 7.

6 Divide 7 by 7: 7 ÷ 7 = 1. Done!

7 784 = 2 × 2 × 2 × 2 × 7 × 7 = 2² × 2² × 7²

8 Take one from each pair: 2 × 2 × 7 = 28.

√784 = 28

Check: 28 × 28 = 784 ✓

Example 2: Find √6.25 by Long Division

Question: Find the square root of 6.25.

1 Group: whole part = 6, decimal part = 25.

2 Largest n² ≤ 6 is 2² = 4. Quotient = 2.

3 Subtract: 6 − 4 = 2. Bring down 25 → 225.

4 Double quotient: 2 × 2 = 4. Write 4_.

5 Find digit: 45 × 5 = 225. It fits!

6 Subtract: 225 − 225 = 0. Done!

7 Place decimal: answer = 2.5.

√6.25 = 2.5

Check: 2.5 × 2.5 = 6.25 ✓

Example 3: Make 252 a Perfect Square

Question: Find the smallest number to multiply 252 to make it a perfect square.

1 Find prime factors of 252.

2 252 = 2 × 2 × 3 × 3 × 7.

3 252 = 2² × 3² × 7.

4 The factor 7 has no pair.

5 Multiply by 7 to give it a pair.

6 252 × 7 = 1764 = 2² × 3² × 7².

7 √1764 = 2 × 3 × 7 = 42.

Multiply by 7. Then √1764 = 42.

Rule: unpaired prime needs a partner!

Term	What It Means	Key Fact
Perfect Square	A number times itself	1, 4, 9, 16, 25...
Square Root	Opposite of squaring	√25 = 5
Ends in	Last digits of squares	0, 1, 4, 5, 6, 9
Odd Number Sum	Sum of first n odd	= n²
Pythagorean Triplet	a² + b² = c²	3, 4, 5
Repeated Subtraction	Subtract 1, 3, 5, 7...	Count steps to 0
Prime Factorisation	Make pairs of primes	Take one from each pair
Long Division	Group in pairs, divide	Works for big numbers
Decimal Square Root	Pair from decimal point	√6.25 = 2.5
Estimating	Between two squares	√50 is near 7

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. 7² = _________.
2. √144 = _________.
3. A perfect square never ends in _________ or _________.
4. Sum of first 5 odd numbers = _________.
5. In the triplet (3, 4, 5), 3² + 4² = _________.

✓ B. True or False

1. 50 is a perfect square. _______
2. √81 = 9. _______
3. A perfect square can end in 7. _______
4. 6, 8, 10 is a Pythagorean triplet. _______
5. √2 is between 1 and 2. _______

✏️ C. Solve These

1. Find √576 by prime factorisation.

Show your work here

2. Find the smallest number to multiply 180 to make it a perfect square.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. √169	a. 7
2. √49	b. 25
3. 5²	c. 13
4. 1 + 3 + 5 + 7 + 9	d. 5²

🎨 E. Find the Square Root

Find √1296 using the long division method. Show each step.

Show your long division work here

✅ Detailed Solutions — Section 5 Handout

A. Fill in the Blanks — Answers

1. 7² = 49. (7 × 7 = 49.)

2. √144 = 12. (12 × 12 = 144.)

3. A perfect square never ends in 2 or 8. (Also not 3 or 7.)

4. Sum of first 5 odd numbers = 25. (1+3+5+7+9 = 25 = 5².)

5. In the triplet (3, 4, 5): 3² + 4² = 9 + 16 = 25 = 5². ✓

B. True or False — Answers

1. 50 is a perfect square. — False. √50 ≈ 7.07. Not a whole number.

2. √81 = 9. — True. 9 × 9 = 81. ✓

3. A perfect square can end in 7. — False. Perfect squares end in 0,1,4,5,6,9. Never 2,3,7,8.

4. 6, 8, 10 is a Pythagorean triplet. — True. 6² + 8² = 36 + 64 = 100 = 10². ✓

5. √2 is between 1 and 2. — True. 1² = 1 < 2 < 4 = 2², so 1 < √2 < 2. ✓

C1. Find √576 by prime factorisation

1 Divide 576 by smallest primes: 576 ÷ 2 = 288.

2 288 ÷ 2 = 144, 144 ÷ 2 = 72, 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9.

3 9 ÷ 3 = 3, 3 ÷ 3 = 1.

4 576 = 2⁶ × 3².

5 √576 = 2³ × 3 = 8 × 3 = 24.

6 Check: 24 × 24 = 576. ✓

√576 = 24

C2. Find smallest number to multiply 180 to make it a perfect square

1 Prime factorisation of 180: 180 = 2² × 3² × 5.

2 For a perfect square every prime must appear in pairs.

3 2² → pair ✓. 3² → pair ✓. 5¹ → unpaired!

4 Multiply by 5 to make 5¹ into 5².

5 180 × 5 = 900 = 2² × 3² × 5² = (2 × 3 × 5)² = 30². ✓

Multiply by 5. New number = 900 = 30².

D. Match the Columns — Answers

1. √169 → c. 13. (13 × 13 = 169.)

2. √49 → a. 7. (7 × 7 = 49.)

3. 5² → b. 25.

4. 1 + 3 + 5 + 7 + 9 → d. 5². (Sum of first 5 odd numbers = 5² = 25.)

E. Find √1296 by Long Division

1 Group digits in pairs from the right: 12 | 96.

2 Find largest integer whose square ≤ 12: 3 (3² = 9 ≤ 12).

3 Quotient so far = 3. Remainder = 12 − 9 = 3. Bring down 96 → 396.

4 Double the quotient: 3 × 2 = 6. New divisor starts with 6_.

5 Find digit d such that (60 + d) × d ≤ 396. Try d = 6: 66 × 6 = 396. ✓

6 Remainder = 396 − 396 = 0. Quotient = 36.

√1296 = 36 (Check: 36 × 36 = 1296. ✓)

🃏 Section 6 — Revision Flashcards

👉

🔢

What is a perfect square?

A number you get by multiplying a whole number by itself.

Examples: 1, 4, 9, 16, 25, 36...

🔎

What digits can a perfect square end in?

A perfect square ends in 0, 1, 4, 5, 6, or 9.

Never ends in 2, 3, 7, or 8.

📈

What is the sum of first n odd numbers?

n²

Example: 1+3+5+7 = 16 = 4²

🔗

What is a Pythagorean triplet?

Three numbers where a² + b² = c².

Example: 3, 4, 5 because 9 + 16 = 25.

❓

How do you find √ by prime factorisation?

Write as primes. Make pairs. Take one from each pair. Multiply them.

💡

How does repeated subtraction work?

Subtract 1, 3, 5, 7... until you reach zero. Count the steps. That count is the square root.

🔧

How do you estimate √50?

49 < 50 < 64, so 7 < √50 < 8.

Closer to 49, so about 7.07.

🎯

How do you find √ of a decimal?

Use long division. Group whole part from right, decimal part from left. Place decimal in answer.

🔢 Class 8 Mathematics — Squares and Square Roots Interactive Visual Revision Notes Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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