Question

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) 190696
(xiii) 586756 (xiv) 27225
(xv)3013696

Answer 1

(i) $\sqrt{441}=\sqrt{3\times 3\times 7\times 7}$

=$3\times 7=21$

(ii) $\sqrt{196}=\sqrt{2\times 2\times 7\times 7}$

=$2\times 7=14$

(iii) $\sqrt{529}=\sqrt{23\times 23}$

=23

(iv) $\sqrt{1764}=\sqrt{2\times 2\times 3\times 3\times 7\times 7}$

= $2\times 3\times 7=42$

(v) $\sqrt{1156}=\sqrt{2\times 2\times 17\times 17}$

=$2\times 17=34$

(vi) $\sqrt{4096}=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}$

= $2\times 2\times 2\times 2\times 2\times 2=64$

(viii) $\sqrt{8281}=\sqrt{7\times 7\times 13\times 13}$

= $7\times 13=91$

(ix) $\sqrt{1}1664=\sqrt{2\times 2\times 2\times 2\times 3\times 3\times 3\times 3\times 3\times 3}$
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= $2\times 2\times 3\times 3\times 3=108$

(x) $\sqrt{47089}=\sqrt{7\times 7\times 31\times 31}$

= $7\times 31=217$

(xi) $\sqrt{24336}=\sqrt{2\times 2\times 2\times 2\times 3\times 3\times 13\times 13}$

= $2\times 2\times 3\times 13=156$

(xii) $\sqrt{190969}=\sqrt{19\times 19\times 23\times 23}$

=$19\times 23=437$

(xiii) $\sqrt{586756}=\sqrt{2\times 2\times 383\times 383}$

=$2\times 383=766$

(xiv) $\sqrt{27225}=\sqrt{3\times 3\times 5\times 5\times 11\times 11}$

=$3\times 5\times 11=165$

(xv) $\sqrt{3013969}=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 7\times 7\times 31\times 31}$

=$2\times 2\times 2\times 7\times 31=1736$