Question

Find the perfect square numbers between 30 and 40.

Answer 1

Hint:
Here, we will use the concept of perfect square to find the solution. The square of any integer is called a perfect square. We will check the squares of the first few positive integers to find which of the perfect squares lie between 30 and 40.

Complete step by step solution:
We know that the square root of a perfect square is always an integer.
We need to find the perfect square numbers between 30 and 40.
We will find the squares of integers to find which of the squares lie in the range 30 to 40.
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 only the square of 6, that is 36, lies between 30 and 40.
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 the square of 7 is 49, the square of all the positive integers greater than 7, is also greater than 49.
Thus, there is no positive integer other than 6 whose square can lie between the numbers 30 and 40.

Hence, we can conclude that 36 is the only perfect square lying between 30 and 40.

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 \[ - 1\].
We need to keep in mind 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.