Perfect squares are the squares of integers
There are 990 integers between 50000 and 60000 that are squares of integers
There are 1000 perfect squares, and 100 perfect cubes between 1 and 1000000
This is because:
[tex]\mathbf{100000 = 1000^2 = 100^3}[/tex]
Rewrite as:
[tex]\mathbf{100000 = x^2 = y^3}[/tex]
Where x lies between 1 and 1000, and y lies between 1 and 100.
If y^3 is a perfect square, then y is a perfect square
List out the perfect squares from 1 to 1000
[tex]\mathbf{y = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100}[/tex]
Take the cube of these numbers
[tex]\mathbf{y^3 = 1^3, 4^3, 9^3, 16^3, 25^3, 36^3, 49^3, 64^3, 81^3, 100^3}[/tex]
[tex]\mathbf{y^3 = 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000}[/tex]
So, the count of numbers that are perfect squares and perfect cubes between 1 and 1000000 is 10.
The count of integers between 50000 and 60000 that are squares of integers is:
[tex]\mathbf{n = x -10}[/tex]
Substitute 1000 for x
[tex]\mathbf{n = 1000 -10}[/tex]
[tex]\mathbf{n = 990}[/tex]
Hence, there are 990 integers between 50000 and 60000 that are squares of integers
Read more about perfect squares at:
https://brainly.com/question/13521012
