A Perfect Square is formed by squaring a whole number. For example 1² = 1; 2² = 4; 3² = 9; 4² = 16; 5² = 25 and so on. Thus 1, 4, 9, 16, 25, etc., are perfect squares. Learn complete information regarding the perfect square and how to find it in this article. We have given examples and also their explanations to understand it easily. Therefore, it is now your part to begin practice and get a complete grip on the concept.
Examples:
1 = 1²; 4 = 2²; 9 = 3²; 16 = 4²; 25 = 5² and so on. Here 1, 4, 9, 16, 25, etc., are perfect squares.
How to Find a Perfect Square or Square Number?
A perfect square number is defined as the product of pairs of equal factors. Or it can also express as grouped in pairs of equal factors.
1. Find out if the following numbers are perfect squares?
(i) 169
(ii) 512
(iii) 64
Solution:
(i) Given number is 169
Find the prime factors of the given number 169.
The prime factors of 169 are 13 and 13.
Grouping the factors into the pairs of equal factors.
(13 × 13)
Factors of the 169 are 13 × 13.
Therefore, 169 is a perfect square.
(ii) Given number is 512
Find the prime factors of the given number 512.
The prime factors of 169 are 8, 8, and 8.
Grouping the factors into the pairs of equal factors.
(8 × 8) × 8
Factors of the 169 are 8 × 8 × 8.
8 is not grouped in pairs of equal factors.
Therefore, 512 is not a perfect square.
(iii) Given number is 64
Find the prime factors of the given number 64.
The prime factors of 64 are 8, and 8.
Grouping the factors into the pairs of equal factors.
(8 × 8)
Factors of the 169 are 8 × 8.
Therefore, 64 is a perfect square.
2. Is 16 a perfect square? If so, find the number whose square is 16.
Solution:
Given number is 16
Find the prime factors of the given number 16.
The prime factors of 16 are 2, 2, 2, and 2.
Grouping the factors into the pairs of equal factors.
(2 × 2) ×(2 × 2)
Therefore, 16 is a perfect square.
Take one number from each group and multiply them to find the number whose square is 16.
2 × 2 = 4.
4 is the number whose square is 16.
3. Is 576 a perfect square? If so, find the number whose square is 576.
Solution:
Given number is 576
Find the prime factors of the given number 16.
The prime factors of 16 are 2, 2, 3, 3, 4, and 4.
Grouping the factors into the pairs of equal factors.
(2 × 2) × (3 × 3) × (4 × 4)
Therefore, 576 is a perfect square.
Take one number from each group and multiply them to find the number whose square is 576.
2 × 3 × 4 = 24.
24 is the number whose square is 576.
4. Show that 288 is not a perfect square.
Solution:
Given number is 288
Find the prime factors of the given number 288.
The prime factors of 16 are 2, 3, 3, 4, and 4.
Grouping the factors into the pairs of equal factors.
2 × (3 × 3) × (4 × 4)
2 is not grouped in pairs of equal factors.
Therefore, 288 is not a perfect square.
5. Find the smallest number by which 100 must be multiplied to make it a perfect square?
Solution:
The given number is 100.
Find the prime factors of the given number 100.
The prime factors of 100 are 5, 5, and 4.
Grouping the factors into the pairs of equal factors.
4 × (5 × 5)
4 is not grouped in pairs of equal factors.
Therefore, by multiplying 4 to 100, we make 100 as a perfect square.
4 is the smallest number by which 100 must be multiplied to make it a perfect square
6. Find the smallest number by which 180 must be divided so as to get a perfect square.
Solution:
The given number is 180.
Find the prime factors of the given number 180.
The prime factors of 180 are 2, 2, 3, 3, and 5.
Grouping the factors into the pairs of equal factors.
5 × (3 × 3) × (2 × 2)
5 is not grouped in pairs of equal factors.
Therefore, by dividing 5 by 180, we make 180 a perfect square.
5 is the smallest number by which 180 must be divided so as to get a perfect square.