(i) 1575
At first,
We’ll resolve the given number into prime factors:
Hence,
1575 = 7 × 25 × 9
= 7 × 3 × 3 × 5 × 5
= (5 × 3) × (5 × 3) × 7
In the above factors only 7 is unpaired
So, in order to get a perfect square the given number should be divided by 7
Hence,
The number whose perfect square is the new number is as following:
= (5 × 3) × (5 × 3)
= (5 × 3) × (5 × 3)
= (5 × 3)2
= (15)2
(ii) 9075
At first,
We’ll resolve the given number into prime factors:
Hence,
9075 = 121 × 25 × 3
= 11 × 11 × 3 × 5 × 5
= (5 × 11) × (5 × 11) × 3
In the above factors only 3 is unpaired
So, in order to get a perfect square the given number should be divided by 3
Hence,
The number whose perfect square is the new number is as following:
=(5 × 11) × (5 × 11)
= (5 × 11)2
= (55)2
(iii) 4851
At first,
We’ll resolve the given number into prime factors:
Hence,
4851 = 11 × 49 × 9
= 11 × 3 × 3 × 7 × 7
= (7 × 3) × (7 × 3) × 11
In the above factors only 11 is unpaired
So, in order to get a perfect square the given number should be divided by 11
Hence,
The number whose perfect square is the new number is as following:
=(7 × 3) × (7 × 3)
= (7 × 3)2
= (21)2
(iv) 3380
At first,
We’ll resolve the given number into prime factors:
Hence,
3380 = 4 × 5 × 169
= 2 × 13 × 13 × 2 × 5
= (2 × 13) × (2 × 13) × 5
In the above factors only 5 is unpaired
So, in order to get a perfect square the given number should be divided by 5
Hence,
The number whose perfect square is the new number is as following:
=(2 × 13) × (2 × 13)
= (2 × 13)2
= (26)2
(v) 4500
At first,
We’ll resolve the given number into prime factors:
Hence,
4500 = 4 × 125 × 9
= 2 × 2 × 3 × 3 × 5 × 5 × 5
= (5 × 3 × 2) × (5 × 3 × 2) × 5
In the above factors only 5 is unpaired
So, in order to get a perfect square the given number should be divided by 5
Hence,
The number whose perfect square is the new number is as following:
=(5 × 3 × 2) × (5 × 3 × 2)
= (5 × 2 × 3) × (5 × 2 × 3)
= (5 × 2 × 3)2
= (30)2
(vi) 7776
At first,
We’ll resolve the given number into prime factors:
Hence,
7776 = 32 × 243
= 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 2
= (2 × 2 × 3 × 3) × (2 × 2 × 3 × 3) × 2 × 3
In the above factors only 2 and 3 are unpaired
So, in order to get a perfect square the given number should be divided by 6
Hence,
The number whose perfect square is the new number is as following:
= (2 × 2 × 3 × 3) × (2 × 2 × 3 × 3)
= (2 × 2 × 3 × 3)2
= (36)2
(vii) 8820
At first,
We’ll resolve the given number into prime factors:
Hence,
8820 = 4 × 5 × 9 × 49
= 2 × 2 × 3 × 3 × 7 × 7 × 5
= (7 × 3 × 2) × (7 × 3 × 2) × 5
In the above factors only 5 is unpaired
So, in order to get a perfect square the given number should be divided by 5
Hence,
The number whose perfect square is the new number is as following:
=(7 × 3 × 2) × (7 × 3 × 2)
= (7 × 3 × 2)2
= (42)2
(viii) 4056
At first,
We’ll resolve the given number into prime factors:
Hence,
4056 = 8 × 3 × 169
= 2 × 2 × 13 × 13 × 3 × 2
= (13 × 2) × (13 × 2) × 6
In the above factors only 6 is unpaired
So, in order to get a perfect square, the given number should be divided by 6
Hence,
The number whose perfect square is the new number is as following:
=(13 × 2) × (13 × 2)
= (13 × 2)2
= (26)2
1575 = 3 x 3 x 5 x 5 x 7
Grouping them into pairs of equal factors:
1575 = (3 x 3) x (5 x 5) x 7
The factor, 7 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 1575 must be divided by 7 for it to be a perfect square.
The new number would be (3 x 3) x (5 x 5).
Furthermore, we have:
(3 x 3) x (5 x 5) = (3 x 5) x (3 x 5)
Hence, the number whose square is the new number is:
3 x 5 = 15
Here, prime factor 7 has no pair. Therefore 252 must be multiplied by 7 to make it a perfect square.
\therefore252\times7=1764
And (i) \sqrt{1764}=2\times3\times7=42
(ii) 180 = 2 x 2 x 3 x 3 x 5
Here, prime factor 5 has no pair. Therefore 180 must be multiplied by 5 to make it a perfect square.
\therefore180\times5=900
And \sqrt{900}=2\times3\times5=30
(iii) 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7
Here, prime factor 7 has no pair. Therefore 1008 must be multiplied by 7 to make it a perfect square.
\therefore1008\times7=7056
And \sqrt{7056}=2\times2\times3\times7=84
(iv) 2028 = 2 x 2 x 3 x 13 x 13
Here, prime factor 3 has no pair. Therefore 2028 must be multiplied by 3 to make it a perfect square.
\therefore2028\times3=6084
And \sqrt{6084}=2\times2\times3\times3\times13\times13=78
(v) 1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3
Here, prime factor 2 has no pair. Therefore 1458 must be multiplied by 2 to make it a perfect square.