Using the prime factorisation method, find which of the following numbers are perfect squares:

Solution:

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.