Question

2. Show that each of the following numbers is a perfect square. In cach case. find the number
whose square is the given number:
(1) 1225
(11) 2601
(iii) 5929
(iv) 7056
(v) 8281
3. By what least number should the given number be multiplied to get a perfect square
number? In each case, find the number whose square is the new number.
(1) 3675
(11) 2156
(111) 3332
(iv) 2925
(v) 9075
(vi) 7623
(vii) 3380
(viii) 2475
please tell me step by step
please tell me fast please ​

Answer 1

$\huge\underlinde\red{Answer:-}$

1. √1225 = 35
2. √2601 = 51
3. √5929 = 77
4. √7056 = 84
5. √8281 = 91

Solution:-

   (i) Given 1225 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 1225 into prime factors, we get 1225 = 25 X 49

= 5 X 5 X 7 X 7

= 5 X 7 X 5 X 7

= 35 X 35

= $(35)^{2}$

Hence, 35 is the number whose square is 1225

∴ 1225 is a perfect square.

 (ii) Given 2601 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 2601 into prime factors, we get 2601 = 9 X 289 .

= 3 X 3 X 17 X 17  

= 3 X 17 X 3 X 17  

= 51 X 51  

=$(51)^{2}$

 Hence, 51 is the number whose square is 2601

∴ 2601 is a perfect square.

   (ii)  Given 5929 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 5929 into prime factors, we get 5929 = 11 X 539

= 11 X 7 X 77

= 77 X 77

= $(77)^{2}$

 Hence, 51 is the number whose square is 2601

∴ 2601 is a perfect square.

 Given 7056 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 7056 into prime factors, we get 7056 = 11 X 539

= 12 X 588

= 12 X 7 X 84  

= 84 X 84                

= $(84)^{2}$

 Hence, 84 is the number whose square is 5929

∴ 7056 is a perfect square.

(v) Given 8281 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 8281 into prime factors, we get 8281 = 49 X 169

= 7 X 7 X 13 X 13  

= 7 X 13 X 7 X 13  

= 91 X 91

= $(91)^{2}$

Hence, 91 is the number whose square is 8281

∴ 8281 is a perfect square

$\huge\underline\red{Solution:- }$

 New number 11025 is the square of 105.

Step-by-step explanation:

Given number is 3675.

To get perfect square by multiplying least no.

Prime factorisation of 3675 is

3 x 5 x 5 x 7 x 7

or, 3 x 5² x 7²

So, we can see to get a perfect square we need to multiply with minimum 3.

It will give 3² x 5² x 7² which is a perfect square.

So, 3² x 5² x 7² = 11025

Hence, new no. is 11025.

so, the square root of 11025 is 3 x 5 x 7 = 105

So, 11025 is the square of 105.