Find the square root of each of the following by prime factorization.
(i) 441
(ii) 196
(iii) 529
(iv) 1764
(v) 1156
(vi) 4096
(vii) 7056
(viii) 8281
(ix) 11664
(x) 47089
(xi) 24336
(xii) 190969
(xiii) 586756
(xiv) 27225
(xv)3013696.

To find: 

We have to find the square root of each of the given numbers by prime factorization.

Solution:

(i) Prime factorization of 441 is,

$441=3\times3\times7\times7$

$=(3\times7)^2$

Therefore,

$\sqrt{441}=\sqrt{(3\times7)^2}$

$=21$

(ii) Prime factorization of 196 is,

$196=2\times2\times7\times7$

$=(2\times7)^2$

Therefore,

$\sqrt{196}=\sqrt{(2\times7)^2}$

$=14$ 

(iii) Prime factorization of 529 is,

$529=23\times23$

$=(23)^2$

Therefore,

$\sqrt{529}=\sqrt{(23)^2}$

$=23$  

(iv) Prime factorization of 1764 is,

$1764=2\times2\times3\times3\times7\times7$

$=(2\times3\times7)^2$

Therefore,

$\sqrt{1764}=\sqrt{(2\times3\times7)^2}$

$=42$   

(v) Prime factorization of 1156 is,

$1156=2\times2\times17\times17$

$=(2\times17)^2$

Therefore,

$\sqrt{1156}=\sqrt{(2\times17)^2}$

$=34$   

(vi) Prime factorization of 4096 is,

$4096=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2$

$=(2\times2\times2\times2\times2\times2)^2$

Therefore,

$\sqrt{4096}=\sqrt{(2\times2\times2\times2\times2\times2)^2}$

$=64$   

(vii) Prime factorization of 7056 is,

$7056=2\times2\times2\times2\times3\times3\times7\times7$

$=(2\times2\times3\times7)^2$

Therefore,

$\sqrt{7056}=\sqrt{(2\times2\times3\times7)^2}$

$=84$ 

(viii) Prime factorization of 8281 is,

$8281=7\times7\times13\times13$

$=(7\times13)^2$

Therefore,

$\sqrt{8281}=\sqrt{(7\times13)^2}$

$=91$  

(ix) Prime factorization of 11664 is,

$11664=2\times2\times2\times2\times3\times3\times3\times3\times3\times3$

$=(2\times2\times3\times3\times3)^2$

Therefore,

$\sqrt{11664}=\sqrt{(2\times2\times3\times3\times3)^2}$

$=108$   

(x) Prime factorization of 47089 is,

$47089=7\times7\times31\times31$

$=(7\times31)^2$

Therefore,

$\sqrt{47089}=\sqrt{(7\times31)^2}$

$=217$    

(xi) Prime factorization of 24336 is,

$24336=2\times2\times2\times2\times3\times3\times13\times13$

$=(2\times2\times3\times13)^2$

Therefore,

$\sqrt{24336}=\sqrt{(2\times2\times3\times13)^2}$

$=156$    

(xii) Prime factorization of 190969 is,

$190969=19\times19\times23\times23$

$=(19\times23)^2$

Therefore,

$\sqrt{190969}=\sqrt{(19\times23)^2}$

$=437$    

(xiii) Prime factorization of 586756 is,

$586756=2\times2\times383\times383$

$=(2\times383)^2$

Therefore,

$\sqrt{586756}=\sqrt{(2\times383)^2}$

$=766$    

(xiv) Prime factorization of 27225 is,

$27225=3\times3\times5\times5\times11\times11$

$=(3\times5\times11)^2$

Therefore,

$\sqrt{27225}=\sqrt{(3\times5\times11)^2}$

$=165$    

(xv) Prime factorization of 3013696 is,

$3013696=2\times2\times2\times2\times2\times2\times7\times7\times31\times31$

$=(2\times2\times2\times7\times31)^2$

Therefore,

$\sqrt{3013696}=\sqrt{(2\times2\times2\times7\times31)^2}$

$=1736$    

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Updated on: 10-Oct-2022

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