Natural numbers that are squares of other natural numbers are called perfect square or square number.
For example;
We know that; 1 = 1²; 4 = 2²; 9 = 3²; 16 = 4²; 25 = 5² and so on.
Thus 1, 4, 9, 16, 25, etc., are perfect squares.
To find out if the given number is a perfect square:
If the prime factors of a number are grouped in pairs of equal factors, then that number is called a perfect square. Or, in other words if a perfect square number is always expressible as the product of pairs of equal factors.
1. Find out if the following numbers are perfect squares:
(i) 144 (ii) 90 (iii) 180
(i) 144
Resolving 144 into prime factors, we get
144 = 2 × 2 × 2 × 2 × 3 × 3
(grouping the factors into the pairs of equal factors)
Therefore, 144 is a perfect square.
(ii) 90
Resolving 90 into prime factors, we get
90 = 2 × 3 × 3 × 5
(Here 3 is grouped in pairs of equal factors and 2 and 5 are not grouped in pairs of equal factors)
Therefore, 90 is not a perfect square.
(iii) 180
Resolving 180 into prime factors, we get
180 = 2 × 2 × 3 × 3 × 5
(Here 2 and 3 are grouped in pairs of equal factors and 5 is not grouped in pairs of equal factors)
Therefore, 180 is not a perfect square.
2. Is 36 a perfect square? If so, find the number whose square is 36.
Solution:
Resolving 36 into prime factors, we get
36 = 2 × 2 × 3 × 3.
Thus, 36 can be expressed as a product of pairs of equal factors.
Therefore, 36 is a perfect square.
Also, 36 = (2 × 3) × (2 × 3) = (6 × 6) = 6²
Hence, 6 is the number whose square is 36.
3. Is 196 a perfect square? If so, find the number whose square is 196.
Solution:
Resolving 196 into prime factors, we get
196 = 2 x 2 x 7 x 7.
Thus, 196 can be expressed as a product of pairs of equal factors.
Therefore, 196 is a perfect square.
Also, 196 = (2 x 7) x (2 x 7) = (14 x 14) = (14)².
Hence, 14 is the number whose square is 196.
4. Show that 200 is not a perfect square.
Solution:
Resolving 200 into prime factors, we get
200 =2 x 2 x 2 x 5 x 5.
Making pairs of equal factors, we find that 2 is left.
Hence, 200 is not a perfect square.
5. Find the smallest number by which 252 must be multiplied to make it a perfect square.
Solution:
252 = 2 × 2 × 3 × 3 × 7
We observe that 2 and 3 are grouped in pairs and 7 is left unpaired.
If we multiply 252 by the factor 7 then,
252 × 7 = 2 × 2 × 3 × 3 × 7 × 7
1764 = 2 × 2 × 3 × 3 × 7 × 7, which is a perfect square.
Therefore, the required smallest number is 7.
6. Find the smallest number by which 396 must be divided so as to get a perfect square.
Solution:
396 = 2 × 2 × 3 × 3 × 11
We observe that 2 and 3 are grouped in pairs and 11 is left unpaired.
If we divide 396 by the factor 11 then,
396 ÷ 11 = (2 × 2 × 3 × 3 × 1̶1̶)/1̶1̶
= 2 × 2 × 3 × 3 = 36, which is a perfect square.
Therefore, the required smallest number is 11.
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