Question

Using the prime factorisation method, find which of the following numbers are perfect squares:
(i) 441
(ii) 576
(iii) 11025
(iv) 1176
(v) 5625
(vi) 9075
(vii) 4225
(viii) 1089

Answer 1

A perfect square can always be expressed as a product of equal factors.

(i)
Resolving into prime factors:
$441=49\times 9=7\times 7\times 3\times 3=7\times 3\times 7\times 3=21\times 21=(21{)}^2$

Thus, 441 is a perfect square.

(ii)
Resolving into prime factors:
$576=64\times 9=8\times 8\times 3\times 3=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3=24\times 24=(24{)}^2$

Thus, 576 is a perfect square.

(iii)
Resolving into prime factors:
$11025=441\times 25=49\times 9\times 5\times 5=7\times 7\times 3\times 3\times 5\times 5=7\times 5\times 3\times 7\times 5\times 3=105\times 105=(105{)}^2$

Thus, 11025 is a perfect square.

(iv)
Resolving into prime factors:
$1176=7\times 168=7\times 21\times 8=7\times 7\times 3\times 2\times 2\times 2$

1176 cannot be expressed as a product of two equal numbers. Thus, 1176 is not a perfect square.

(v)
Resolving into prime factors:
$5625=225\times 25=9\times 25\times 25=3\times 3\times 5\times 5\times 5\times 5=3\times 5\times 5\times 3\times 5\times 5=75\times 75=(75{)}^2$

Thus, 5625 is a perfect square.

(vi)
Resolving into prime factors:
$9075=25\times 363=5\times 5\times 3\times 11\times 11=55\times 55\times 3$

9075 is not a product of two equal numbers. Thus, 9075 is not a perfect square.

(vii)
Resolving into prime factors:
$4225=25\times 169=5\times 5\times 13\times 13=5\times 13\times 5\times 13=65\times 65=(65{)}^2$

Thus, 4225 is a perfect square.

(viii)
Resolving into prime factors:
$1089=9\times 121=3\times 3\times 11\times 11=3\times 11\times 3\times 11=33\times 33=(33{)}^2$

Thus, 1089 is a perfect square.