CBSE Class 8 Mathematics Worksheet - Square and Square Roots | PDF - Free Printable

Let's solve each question from the Maths Worksheet (Class VIII) - Chapter 6: Square and Square Roots step by step with clear explanations.

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1. Which of the following numbers are perfect squares?

(a) 3364
(b) 4489
(c) 4358

Solution:
A perfect square is a number that can be expressed as the square of an integer.

- (a) 3364:
Try to find the square root:
$ \sqrt{3364} = 58 $, since $ 58^2 = 3364 $. ✔ Perfect square

- (b) 4489:
$ \sqrt{4489} = 67 $, because $ 67^2 = 4489 $. ✔ Perfect square

- (c) 4358:
Check if it’s a perfect square:
$ 66^2 = 4356 $, $ 67^2 = 4489 $.
4358 lies between them and not equal to any square. ✘ Not a perfect square

✔ Answer: (a) and (b) are perfect squares.

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2. Using the property of squares, find the value of the following.

We use the identity:
$ a^2 - b^2 = (a - b)(a + b) $

(a) $ 24^2 - 23^2 $
= $ (24 - 23)(24 + 23) = (1)(47) = 47 $

(b) $ 50^2 - 49^2 $
= $ (50 - 49)(50 + 49) = (1)(99) = 99 $

(c) $ 105^2 - 104^2 $
= $ (105 - 104)(105 + 104) = (1)(209) = 209 $

✔ Answers:
(a) 47
(b) 99
(c) 209

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3. Without adding, find the sum.

This uses the formula for the sum of first $ n $ odd numbers:
$ 1 + 3 + 5 + \ldots + (2n - 1) = n^2 $

(a) $ 1 + 3 + 5 + 7 + 9 $
These are the first 5 odd numbers → Sum = $ 5^2 = 25 $

(b) $ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 $
Total terms: From 1 to 23 (odd), so $ \frac{23 + 1}{2} = 12 $ terms
Sum = $ 12^2 = 144 $

✔ Answers:
(a) 25
(b) 144

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4. Find the square root of the following by prime factorisation method.

(a) 1024

Factorise 1024:
$$
1024 ÷ 2 = 512 \\
512 ÷ 2 = 256 \\
256 ÷ 2 = 128 \\
128 ÷ 2 = 64 \\
64 ÷ 2 = 32 \\
32 ÷ 2 = 16 \\
16 ÷ 2 = 8 \\
8 ÷ 2 = 4 \\
4 ÷ 2 = 2 \\
2 ÷ 2 = 1
$$
So, $ 1024 = 2^{10} $

Now, $ \sqrt{1024} = \sqrt{2^{10}} = 2^{5} = 32 $

✔ Answer: 32

(b) 8836

Let’s factorise 8836:

Check divisibility:
- Even → divide by 2:
$ 8836 ÷ 2 = 4418 $
$ 4418 ÷ 2 = 2209 $

Now, 2209 — check if divisible by small primes:
Try $ 47^2 = 2209 $?
$ 47 × 47 = (50 - 3)^2 = 2500 - 300 + 9 = 2209 $ ✔ Yes!

So, $ 8836 = 2^2 × 47^2 $

Then, $ \sqrt{8836} = 2 × 47 = 94 $

✔ Answer: 94

(c) 8464

Factorise 8464:

Even → divide by 2:
$ 8464 ÷ 2 = 4232 $
$ 4232 ÷ 2 = 2116 $
$ 2116 ÷ 2 = 1058 $
$ 1058 ÷ 2 = 529 $

Now, 529 → $ 23^2 = 529 $

So, $ 8464 = 2^4 × 23^2 $

Then, $ \sqrt{8464} = 2^2 × 23 = 4 × 23 = 92 $

✔ Answer: 92

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5. Find the smallest number by which 2475 must be multiplied to get a perfect square.

First, prime factorise 2475:

- 2475 is odd → not divisible by 2
- Divisible by 5:
$ 2475 ÷ 5 = 495 $
$ 495 ÷ 5 = 99 $
$ 99 ÷ 3 = 33 $
$ 33 ÷ 3 = 11 $
$ 11 ÷ 11 = 1 $

So, $ 2475 = 3^2 × 5^2 × 11^1 $

For a perfect square, all exponents must be even.

Here, only 11 has an odd exponent (1). So multiply by 11 to make it $ 11^2 $

✔ Answer: 11

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6. 4096 soldiers are arranged in an auditorium such that there are as many soldiers in a row as there are rows. How many rows are there?

Let number of rows = $ x $, then number of soldiers per row = $ x $

So total soldiers = $ x × x = x^2 $

Given: $ x^2 = 4096 $

Find $ x = \sqrt{4096} $

We know:
- $ 64^2 = 4096 $ (since $ 60^2 = 3600 $, $ 64^2 = (60+4)^2 = 3600 + 480 + 16 = 4096 $)

✔ Answer: 64 rows

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7. Find the least perfect square exactly divisible by each one of the numbers 4, 5, 10.

We need LCM of 4, 5, 10, and then make it a perfect square.

Prime factors:
- 4 = $ 2^2 $
- 5 = $ 5^1 $
- 10 = $ 2 × 5 $

LCM = $ 2^2 × 5 = 20 $

But 20 is not a perfect square. To make it a perfect square, we need even powers.

Currently: $ 2^2 × 5^1 $ → 5 has odd power.

Multiply by 5 → $ 2^2 × 5^2 = 100 $

Check: $ \sqrt{100} = 10 $, and 100 divisible by 4, 5, 10?

- 100 ÷ 4 = 25 ✔
- 100 ÷ 5 = 20 ✔
- 100 ÷ 10 = 10 ✔

✔ Answer: 100

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8. Find the least square number exactly divisible by each one of the numbers 6, 9, 10, 15, and 20.

Step 1: Find LCM of 6, 9, 10, 15, 20

Prime factorise:
- 6 = $ 2 × 3 $
- 9 = $ 3^2 $
- 10 = $ 2 × 5 $
- 15 = $ 3 × 5 $
- 20 = $ 2^2 × 5 $

Take highest powers:
- $ 2^2 $ (from 20)
- $ 3^2 $ (from 9)
- $ 5^1 $ (from 10, 15, 20)

So LCM = $ 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180 $

Now, 180 is not a perfect square: $ 180 = 2^2 × 3^2 × 5^1 $

To make it a perfect square, we need even powers → multiply by 5

New number: $ 180 × 5 = 900 $

Check: $ \sqrt{900} = 30 $, perfect square.

Is 900 divisible by all?

- 900 ÷ 6 = 150 ✔
- 900 ÷ 9 = 100 ✔
- 900 ÷ 10 = 90 ✔
- 900 ÷ 15 = 60 ✔
- 900 ÷ 20 = 45 ✔

✔ Answer: 900

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✔ Final Answers Summary:

1. Perfect squares: (a) 3364, (b) 4489
2. (a) 47, (b) 99, (c) 209
3. (a) 25, (b) 144
4. (a) 32, (b) 94, (c) 92
5. 11
6. 64 rows
7. 100
8. 900

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