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

(i) 441 (ii) 576


(iii) 11025 (iv) 1176


(v) 5625 (vi) 9075


(vii) 4225 (viii) 1089

(i) 441


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


441 = 49 × 9


= 7 × 7 × 3 × 3


= (7 × 3) × (7 × 3)


= 21 × 21


= (21)2


Hence, it is a perfect square.


(ii) 576


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


576 = 64 × 9


= 8 × 8 × 3 × 3


= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3


= (2 × 2 × 2 × 3) × (2 × 2 × 2 × 3)


= 24 × 24


= (24)2


Hence, it is a perfect square.


(iii) 11025


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


11025 = 441 × 25


= 49 × 9 × 5 × 5


= 7 × 7 × 3 × 3 × 3 × 3 × 5 × 5


= (7 × 5 × 3 × 3) × (7 × 5 × 3 × 3)


= 24 × 24


= (24)2


Hence,


It is a perfect square.


(iv) 1176


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


1176 = 7 × 168


= 7 × 8 × 21


= 7 × 2 × 2 × 2 × 7 × 3


Hence,


We can see that,


The number 1176 cannot be expressed as a product of two equal numbers.


Thus,


1176 is not a perfect square.


(v) 5625


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


5625 = 225 × 25


= 9 × 25 × 25


= 5 × 5 × 5 × 5 × 3 × 3


= (5 × 5 × 3) × (5 × 5 × 3)


= 75 × 75


= (75)2


Hence,


It is a perfect square.


(vi) 9075


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


9075 = 25 × 363


= 25 × 3 × 121


= 5 × 5 × 3 × 11 × 11


= 25 × 3 × 121


Hence,


We can see that,


The number 9075 cannot be expressed as a product of two equal numbers.


Thus,


9075 is not a perfect square.


(vii) 4225


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


4225 = 25 × 169


= 5 × 5 × 13 × 13


= (5 × 13) × (5 × 13)


= 65 × 65


= (65)2


Hence,


It is a perfect square.


(viii) 1089


In order to find if the given number is a perfect square,


At first,


We’ll resolve the given number into prime factors:


Hence,


1089 = 121 × 9


= 11 × 11 × 3 × 3


= 11 × 11 × 3 × 3


= (11 × 3) × (11 × 3)


= 33 × 33


= (33)2


Hence,


It is a perfect square.


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