Question

List the first four perfect squares. Next, list the first four perfect cubes. Following that same pattern find a number that is a perfect square and also a perfect cube other than the number 1. Explain how you found your answer.

Answer 1

The first four perfect squares are 1, 4, 9, and 16, and the first four perfect cubes are 1, 8, 27, and 64. A number that is both a perfect square and a cube is 64, which is 2 to the power of 6. Perfect squares and cubes are found by raising integers to the power of 2 for squares and 3 for cubes.

Step-by-step explanation:

Finding Perfect Squares and Cubes

The first four perfect squares are numbers that can be expressed as the square of an integer. These are:

1. 12 = 1
2. 22 = 4
3. 32 = 9
4. 42 = 16

The first four perfect cubes are numbers that can be expressed as the cube of an integer. These are:

1. 13 = 1
2. 23 = 8
3. 33 = 27
4. 43 = 64

A number that is both a perfect square and a perfect cube, other than the number 1, can be found by looking for a power that is both the square and cube of an integer. One such number is 64, which is 82 and also 43. To find more numbers like this, you can look for integers that are powers of 6 (since 2 × 3 = 6), such as 26 = 64, 36 = 729, and so on. These numbers are both perfect squares and perfect cubes because a square of a cube or a cube of a square will result in an exponent that is a multiple of both 2 and 3.

Answer 2

We can list the first four perfect squares:

`1^2=1,2^2=4,3^2=9,4^2=16`

We can following the same pattern to find the rest of the square and compare them to those that are a perfect cube:

`5^2=25,6^2=36,7^2=49,8^2=64,9^2=81,10^2=100`

Then, we have

`1^3=1,2^3=8,3^3=27,4^3=64,5^3=125^{}`

We can see from the two lists:

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100....

Perfect cubes: 1, 8, 27, 64, 125....

Comparing these two lists, we have that the number 64 is, at the same time, a perfect square and a perfect cube.

Then, to find the answer, we need to make two lists, first, raising the natural numbers to the second power, and then, to the third power, and compare when we get a number is a perfect square and a perfect cube (the other apart from 1).