What is the first natural number that is a perfect square
and a perfect cube? Can you find the second? The third?
To find the first natural number that is both a perfect square and a perfect cube, we need to look for a number that has an integer square root and an integer cube root.
We can start by checking the squares of the natural numbers and see if any of them have an integer cube root. We can begin with the square of 1, which is 1. The cube root of 1 is also 1, so 1 is a number that meets the criteria.
For the second number, we can continue checking the squares of the natural numbers. The square of 2 is 4, but the cube root of 4 is not an integer. We can move on to the square of 3, which is 9. The cube root of 9 is 3, so 9 is the second number that meets the criteria.
To find the third number, we can continue this process by checking the squares of the natural numbers. The square of 4 is 16, but the cube root of 16 is not an integer. The square of 5 is 25, and the cube root of 25 is also 25, making 25 the third number that meets the criteria.
So, the first three natural numbers that are both perfect squares and perfect cubes are 1, 9, and 25.
squares: 1 4 9
cubes: 1 8 27