To check whether the given natural number is a perfect square or not, we can follow the below steps:
1. Observe the given natural number.
2. Then write the given natural number as a product of prime factors.
3. Now group the factors in pairs such a way that both factors in each pair are equal.
4. Now see whether some factors are leftover or not. If no factor is left over in grouping, then the given number is a perfect square. Otherwise, it is not a perfect square.
5. Take one factor from each group and multiply them to obtain the number whose square is the given number.
Before we go to this concept, let's first recall about prime numbers and prime factors.
The numbers, which can be divided only by itself and \(1\) are called prime numbers.
For example: \(2, 3, 5, 7, 11, 13, 17, 19\) etc.
The given number could have many factors, but we have to divide that number repeatedly by the prime number, which is also the factor of the same number.
Let's see an example to understand this concept clear.
I) Find the number whose square is 256.
Step I) The given number is 256
Step II) Now we have to write 256 as a product of prime factors.
Step III) Now we can group the prime factors.
That is .
Here no factor is left over in grouping; then the given number is a perfect square.
Step IV) Now take one factor commonly from each group.
\(=\) . If we multiply this, we can get the required answer that is 16.
The given number 256 is a perfect square.
Therefore the square of 16 is 256.
Multiply or divide to obtain perfect square:
Not all squares are perfect squares, but if we multiply or divide some number with a specific number, we can obtain the perfect square.
Do the prime factorization on 200.
Now we can group the prime factors that, . Here you can see that number \(2\) is ungrouped or unpaired.
It means that the number 200 is not a perfect square.
If we multiply or divide the 200 by \(2\), then we can obtain the perfect square.
If we multiply \(=\) 400. Here 400 is the square of \(20\), which is the perfect square.
If we divide \(=\) 100. Here 100 is the square of \(10\), which is the perfect square.
Therefore applying this method, we can find the perfect square even if the given number is not the perfect square.