Question

How do you determine if a number is a perfect square: 15?

Answer 1

Hint: Here, we need to check whether 15 is a perfect square or not. We will find the squares of the first few integers. If the number 15 lies between the squares of two consecutive integers, then it is not a perfect square. A square of a number is defined as the number obtained by multiplying the number by itself.

Complete step-by-step solution:
The square of any integer is called a perfect square.
This means that the square root of a perfect square is always an integer.
We will find the squares of the first few integers.
The numbers 1, 2, 3, 4, 5, 6, 7, 8 are the first 8 positive integers.
The square of 1 is 1.
The square of 2 is 4.
The square of 3 is 9.
The square of 4 is 16.
The square of 5 is 25.
The square of 6 is 36.
The square of 7 is 49.
The square of 8 is 64.
We can observe that 15 lies between 9 and 16, that is between the square of 3 and 4.
We know that if \[x\] and \[y\] are two positive integers such that \[x > y\], then the square of \[x\] is always greater than the square of \[y\].
Therefore, since 15 lies between the square of 3 and 4, there is no number greater than 4 whose square is 15.

Thus, 15 is not a perfect square because it lies between the squares of two consecutive integers.

Note:
We used the term ‘integer’ in the solution. An integer is a rational number that is not a fraction. For example: 1, \[ - 1\], 3, \[ - 7\], are integers. Integers can be positive like 1, 3, etc. or negative like \[ - 1\].
You should remember that the perfect squares are the squares of integers and not the squares of any real number. The square of \[5.5\] is \[30.25\], which also lies between 30 and 40. However, since \[5.5\] is not an integer, \[30.25\] is not a perfect square.
If the square of a number has 5 in the unit’s place, then it always has 2 in the ten’s place. Since the given number 15 has 5 in the unit’s place but 1 in the ten’s place, it is not a perfect square.