Question

Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:
(i) 1225
(ii) 2601
(iii) 5929
(iv) 7056
(v) 8281

Answer 1

(i) $1225=5\times 5\times 7\times 7$

=$(5{)}^7\times (7{)}^2=(5\times 7{)}^2$
=$(35{)}^2$
Hence it is a perfect square and it is the square of 35
(ii) $2601=3\times 3\times 17\times 17$

=$(3{)}^2\times (17{)}^2=(3\times 17{)}^2=(51{)}^2$
Hence it is a perfect square and it is the squae of 56
(iii) 5929= $7\times 7\times 11\times 11$

= $(7{)}^2\times (11{)}^2=(7times11{)}^2=(77{)}^2$
It is a perfect square and is the square of 77
(iv) 7056= $2\times 2\times 2\times 2\times 3\times 3\times 7\times 7$

=$(2{)}^2\times (2{)}^2\times (3{)}^2\times (7{)}^2$
=$(2\times 2\times 3\times 7{)}^2=(84{)}^2$
$\therefore$ It is a perfect square and is of the square of 84
(v) 8281 = $7\times 7\times 13\times 13$

=$(7{)}^2\times (13{)}^2$
=$(7\times 13{)}^2=(91{)}^2$
$\therefore$ It is a perfect square and is of the square of 91.