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Find the square root of each of the following numbers by using the method of prime factorization:

(a) 121

(b) 441

(c) 625

(d) 729

(e) 1521

(f) 2025

(g) 4096

(h) 5776

(i) 8100

(j) 9216

(k) 11236

(l) 15876

(m) 18496

(a) 121 = 11 x 11

$\sqrt {121} =11$

(b) 441= 3 x 3 x 7 x 7

$\sqrt {441} =3 \times 7=21$

(c) 625= 5 x 5 x 5 x 5

$\sqrt {625} =5 \times 5=25$

(d) 729= 3 x 3 x 3 x 3 x 3 x 3

$\sqrt {729} =3 \times 3 \times 3=27$

(e)1521= 3 x 3 x 13 x 13

$\sqrt {1521} =3 \times 13=39$

(f) 2025= 3 x 3 x 3 x 3 x 5 x 5

$\sqrt {2025} =3 \times 3 \times 5=45$

(g) 4096=2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

$\sqrt {4096} =2 \times 2 \times 2 \times 2 \times 2 \times 2=64$

(h) 5776= 2 x 2 x 2 x 2 x 19 x 19

$\sqrt {5776} =2 \times 2 \times 19=76$

(i) 8100= 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5

$\sqrt {8100} =2 \times 3 \times 3 \times 5=90$

(j) 9216= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3

$\sqrt {9216} =2 \times 2 \times 2 \times 2 \times 2 \times 3=96$

(k)11236 =2 x 2 x 53 x 53

$\sqrt {11236} =2 \times 53=106$

(l) 15876= 2 x 2 x 3 x 3 x 3 x 3 x 7 x 7

$\sqrt {15876} =2 \times 3 \times 3 \times 7=126$

(m) 18496=2 x 2 x 2 x 2 x 2 x 2 x 17 x 17

$\sqrt {18496} =2 \times 2 \times 2 \times 17=136$

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

- 1008
- 1280
- 1875

(i)1008= 2 x 2 x 2 x 2 x 3 x 3 x 7

We can see number 7 is not in pair, So to make a perfect square, it has to be multiplied by 7

Then number will become=7056

Now

7056= 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7

$\sqrt {7056} =2 \times 2 \times 3 \times 7=84$

(ii)1280=2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5

We can see number 5 is not in pair, So to make a perfect square, it has to be multiplied by 5

Then number will become=6400

Now

6400= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5

$\sqrt {6400} =2 \times 2 \times 2 \times 2 \times 5=80$

(iii)1875= 3 x 5 x 5 x 5 x 5

We can see number 3 is not in pair, So to make a perfect square, it has to be multiplied by 3

Then number will become=5625

Now

5625= 3 x 3 x 5 x 5 x 5 x 5

$\sqrt {5625} =3 \times 5 \times 5 =75$

676 students are to be sit in a hall in such a way that each row contains as many students as the number of rows. Find the number of rows and the number of students in each row.

Here we need to find the square root of the Number 676

676=2 x 2 x 13 x 13

$\sqrt {676} =2 \times 13 =26$

So There are 26 rows and each rows has 26 students

What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

(i) 1801

(ii) 856

(iii) 1008001

(iv) 6577525

(i)Last digit is 1 , So one digit can be 1 or 9 as $1^2=1$ and $9^2=81$

(ii)Last digit is 5 , So one digit can be 4 or 6 as $4^2=16$ and $6^2=36$

(i)Last digit is 1 , So one digit can be 1 or 9 as $1^2=1$ and $9^2=81$

(i)Last digit is 5 , So one digit will be 5 as $5^2=25$

The students of a class arranged a gift for the class teacher. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs 1521, find the strength of the class.

Here we need to find the square root of the Number 1521

1521= 3 x 3 x 13 x 13

$\sqrt {1521} =3 \times 13=39$

So There are 39 students and each contributed has Rs 39

Find the least number which when added to 4529 to make it a perfect square?

Let us find the square root of 4529 using Long division method

So remainder is 40

Therfore $67^2 < 4529$

Next perfect square would be $68^2=4624$

hence the number to be added = 4624 - 4529 = 95

So addition of 95 to 4529 will make it perfect square

Find the least number which must be subtracted from 2361 to make it a perfect square?

Let us find the square root of 2361 using Long division method

So remainder is 57

Therfore $48^2 < 2361$

Now if we subtract the remainder from main number, it will be perfect square
So subtraction of 57 from 2361 will make it perfect square

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

(i)600

(ii)2904

(i) 600= 2 x 2 x 2 x 3 x 5 x 5

We can see number 2 and 3 is not in pair, So to make a perfect square, it has to be divided by 6

Then number will become=100

Now

100= 2 x 2 x 5 x 5

$\sqrt {100} =2 \times 5 =10$

(ii)2904=2 x 2 x 2 x 3 x 11 x 11

We can see number 2 and 3 is not in pair, So to make a perfect square, it has to be divided by 6

Then number will become=484

Now

484= 2 x 2 x 11 x 11

$\sqrt {484} =2 \times 11 =22$

Find the value of

$ \sqrt {176 + \sqrt {2401}}$

$ \sqrt {176 + \sqrt {2401}} = \sqrt {176 + 49} = \sqrt {225} =15$

Find the square root

(i).0151129

(ii) 83.3569

Square roots for decimal are found using the same long division method. We put bars on both integral part and decimal part.For integral we move fron the unit's place close to the decimal and move towards left. For decimal part, we start from the decimal and move towards right

(i) By long division method

$\sqrt {.0151129}=.123$

(ii) By long division method

$\sqrt {83.3569}=9.13$

(i)There are _________ perfect squares between 1 and 100

(ii) The square of a proper fraction is ______ than to the fraction

(iii) The square of a even number is _____

(iv) $\sqrt{4096}$ is ____

(v) The digit at the ones place of $37^2$ is ____

(vi) the least number that must be added to 1500 so as to get a perfect square is ___

(i)8

(ii) smaller

(iii) even

(iv) 64

(v) 9

(vi) 39

(i)The square root of 0.64 is 0.8.

(ii) There is no square number between 70 and 80.

(iii)6292 is not a perfect square

(iv) 1000 is a perfect square.

(v) The product of two perfect squares is a perfect square

(vi)The number of digits in a perfect square is even

(i)True

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

Which of the following is not a perfect square number?

(a) 1156

(b) 4657

(c)4624

(d) 7056

A perfect square can never have the following digit in its ones place

(a) 8

(b) 4

(b) 0

(d) 1

The sum of first n odd natural numbers is

(a)$n^2$

(b) $2n$

(c) $n^2 +1$

(d) $n^2 -1$

$\sqrt {.9} $ is

(a).3

(b) .03

(c).94

(d).33

The area of the square field is 234.09 m

(a)65.2 m

(b)59.6 m

(c)51.2 m

(d)61.2 m

The value of x in $\sqrt {1 + \frac {25}{144}} = 1 + \frac {x}{12}$ is

(a)2

(b) 1

(c) 4

(d) 5

Given that $\sqrt {5625} =75$, the value of $\sqrt {0.5625} + \sqrt {56.25}$ is:

(a) 82.5

(b) 0.75

(c) 8.25

(d) 75.05

Which of the following is a pythogorean triplet?

(a) 2,3,4

(b) 6,8,10

(c) 5,7,9

(d) none of these

13. (b)

14. (a)

15. (a)

16. (c)

17. (d)

$a =\sqrt {234.09} = 15.3$

Perimeter=4a=61.2m

18. (b)

$\sqrt {1 + \frac {25}{144}} = 1 + \frac {x}{12}$

$\sqrt { \frac {169}{144}}= 1 + \frac {x}{12}$

$\frac {13}{12} =1 + \frac {x}{12}$

$x=1$

19 . (c)

if $\sqrt {5625} =75$, then

$\sqrt {0.5625} + \sqrt {56.25}= .75 + 7.5= 8.25$

20. (b)

(p) -> (iv)

(q) -> (i)

(r) -> (iii)

(s) -> (ii)

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