Squares and Square Roots (Class 8) | PDF - Free Printable

Let's solve each problem step-by-step with clear explanations.

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1. Find the square root of the following by means of factors

#### i) 529

We will find the prime factorization of 529.

- Try dividing by small primes:
- 529 ÷ 7 = not divisible
- 529 ÷ 11 = not divisible
- 529 ÷ 13 = not divisible
- 529 ÷ 23 = 23 (since 23 × 23 = 529)

So,
$$
529 = 23 \times 23 = 23^2
$$

Therefore,
$$
\sqrt{529} = 23
$$

✔ Answer: 23

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#### ii) 298116

We'll find the prime factorization.

Start dividing:

- It’s even → divide by 2:
$$
298116 ÷ 2 = 149058
$$
$$
149058 ÷ 2 = 74529
$$

Now 74529 is odd. Check divisibility by 3:
Sum of digits: 7+4+5+2+9 = 27 → divisible by 3
$$
74529 ÷ 3 = 24843
$$
$$
24843 ÷ 3 = 8281
$$

Now check 8281:
Try dividing by 7: 8281 ÷ 7 ≈ 1183 → 7×1183 = 8281?
7×1180 = 8260 → 8281 – 8260 = 21 → 7×3=21 → so yes:
$$
8281 ÷ 7 = 1183
$$
Now 1183 ÷ 7 = 169 (since 7×169 = 1183)
And 169 = 13 × 13

So putting it all together:

$$
298116 = 2^2 × 3^2 × 7^2 × 13^2
$$

Now take square root:
$$
\sqrt{298116} = 2 × 3 × 7 × 13 = 546
$$

✔ Answer: 546

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2. Find the smallest number by which 252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.

Step 1: Prime factorize 252

- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1

So,
$$
252 = 2^2 × 3^2 × 7^1
$$

For a perfect square, all exponents must be even.

Here, exponent of 7 is 1 → needs one more 7.

So, multiply by 7.

New number:
$$
252 × 7 = 1764
$$

Now,
$$
1764 = 2^2 × 3^2 × 7^2 = (2 × 3 × 7)^2 = 42^2
$$

So,
$$
\sqrt{1764} = 42
$$

✔ Answer: Multiply by 7; Perfect square = 1764; Square root = 42

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3. Find the smallest number by which 2925 must be divided to get a perfect square. Also, find the square root of the perfect square so obtained.

Factorize 2925:

- 2925 ÷ 5 = 585
- 585 ÷ 5 = 117
- 117 ÷ 3 = 39
- 39 ÷ 3 = 13
- 13 ÷ 13 = 1

So,
$$
2925 = 5^2 × 3^2 × 13^1
$$

Only 13 has an odd exponent (1), so we need to divide by 13.

Divide:
$$
2925 ÷ 13 = 225
$$

Now,
$$
225 = 5^2 × 3^2 = (5 × 3)^2 = 15^2
$$

So,
$$
\sqrt{225} = 15
$$

✔ Answer: Divide by 13; Perfect square = 225; Square root = 15

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4. Find the least square number exactly divisible by each one of the numbers 6, 9, 15, and 20.

We need the least perfect square divisible by 6, 9, 15, 20.

This is equivalent to finding the LCM of these numbers, then making it a perfect square.

First, find LCM of 6, 9, 15, 20.

Prime factorizations:
- 6 = 2 × 3
- 9 = 3²
- 15 = 3 × 5
- 20 = 2² × 5

Take highest powers:
- 2², 3², 5¹ → So LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

But 180 is not a perfect square (exponent of 5 is 1).

To make it a perfect square, make all exponents even.

So multiply by 5 → now 5².

So required number:
$$
2^2 × 3^2 × 5^2 = (2 × 3 × 5)^2 = 30^2 = 900
$$

✔ Answer: 900

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5. Find the least square number exactly divisible by each one of the numbers 8, 12, 15, 20.

Again, find LCM of 8, 12, 15, 20.

Factorizations:
- 8 = 2³
- 12 = 2² × 3
- 15 = 3 × 5
- 20 = 2² × 5

Highest powers:
- 2³, 3¹, 5¹ → LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

Now make 120 a perfect square.

Exponents:
- 2³ → need one more 2 → make 2⁴
- 3¹ → need one more 3 → 3²
- 5¹ → need one more 5 → 5²

So multiply 120 by (2 × 3 × 5) = 30 → gives 120 × 30 = 3600

Check:
$$
3600 = 2^4 × 3^2 × 5^2 = (2^2 × 3 × 5)^2 = (4 × 3 × 5)^2 = 60^2
$$

✔ Answer: 3600

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6. Find the square root of:

#### (i) 9126441

Use long division method or factorization.

Try estimating:

- √9126441 → around 3000? Try 3000² = 9,000,000
- 3010² = (3000 + 10)² = 9,000,000 + 2×3000×10 + 100 = 9,000,000 + 60,000 + 100 = 9,060,100 → too big
- Try 3015² = ?

Alternatively, use prime factorization or try dividing.

But let's do long division-style.

Actually, let's try:

Try 3017²:

Or better, note that 3017² = ?

Wait, let's try:

Try 3017²:
- 3000² = 9,000,000
- 17² = 289
- 2×3000×17 = 102,000
- So (3000+17)² = 9,000,000 + 102,000 + 289 = 9,102,289 → too big

Wait, our number is 9,126,441

Try 3020² = (3000 + 20)² = 9,000,000 + 2×3000×20 + 400 = 9,000,000 + 120,000 + 400 = 9,120,400

Now 9,126,441 – 9,120,400 = 6,041

Try 3021² = 3020² + 2×3020 + 1 = 9,120,400 + 6,040 + 1 = 9,126,441 ✔

So,
$$
\sqrt{9126441} = 3021
$$

✔ Answer: 3021

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#### (ii) 63409369

Estimate:

- √63409369 → ~ 7960? Because 8000² = 64,000,000
- 7900² = 62,410,000
- 7960² = ?

Try 7960²:
= (8000 – 40)² = 8000² – 2×8000×40 + 40² = 64,000,000 – 640,000 + 1,600 = 63,361,600

Now 63,409,369 – 63,361,600 = 47,769

Try 7970² = (7960 + 10)² = 7960² + 2×7960×10 + 100 = 63,361,600 + 159,200 + 100 = 63,520,900 → too big

Try 7965²?

Better: Try 7963²?

Wait, try 7963²:

Let’s compute:
- 7960² = 63,361,600
- 7963 = 7960 + 3
- So (7960 + 3)² = 7960² + 2×7960×3 + 9 = 63,361,600 + 47,760 + 9 = 63,409,369 ✔

So,
$$
\sqrt{63409369} = 7963
$$

✔ Answer: 7963

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7. Find the least number that must be subtracted from 7581 to obtain a perfect square. Find the perfect square and its square root.

Find the largest perfect square ≤ 7581.

Try √7581:

- 87² = 7569
- 88² = 7744 → too big

So 87² = 7569

Now,
$$
7581 - 7569 = 12
$$

So subtract 12 to get 7569, which is a perfect square.

$$
\sqrt{7569} = 87
$$

✔ Answer: Subtract 12; Perfect square = 7569; Square root = 87

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## ✔ Final Answers Summary:

1. i) √529 = 23
ii) √298116 = 546

2. Multiply 252 by 7 → perfect square = 1764, √1764 = 42

3. Divide 2925 by 13 → perfect square = 225, √225 = 15

4. Least square divisible by 6,9,15,20 = 900

5. Least square divisible by 8,12,15,20 = 3600

6. i) √9126441 = 3021
ii) √63409369 = 7963

7. Subtract 12 from 7581 → perfect square = 7569, √7569 = 87

Let me know if you'd like any steps explained further!