What pattern do you notice in the numbers below?

Question

1 = 1 × 1, and 1 × 1 × 1, and 1 × 1 × 1
64 = 8 × 8, and 4 × 4 × 4
729 = 27 × 27, and 9 × 9 × 9
4,096 = 64 × 64, and 16 × 16 × 16


They are perfect squares.



They are perfect cubes.



They are both perfect squares and perfect cubes.



They are neither perfect squares nor perfect cubes.

Answer 1

To analyze the numbers provided, let's examine each of them in terms of their square and cube configurations.

• 1:

  • \(1 = 1^2\) (perfect square)
  • \(1 = 1^3\) (perfect cube)

• 64:

  • \(64 = 8^2\) (perfect square)
  • \(64 = 4^3\) (perfect cube)

• 729:

  • \(729 = 27^2\) (not a perfect square, since \(27\) is not an integer base of a square)
  • \(729 = 9^3\) (perfect cube)

• 4,096:

  • \(4,096 = 64^2\) (perfect square)
  • \(4,096 = 16^3\) (perfect cube)

From the analysis, we can summarize:

1. 1 is both a perfect square and a perfect cube.
2. 64 is both a perfect square and a perfect cube.
3. 729 is not a perfect square but is a perfect cube.
4. 4,096 is both a perfect square and a perfect cube.

The majority show either being a perfect square or a perfect cube, with 1 and 64 being both.

Given this understanding, the most inclusive classification is:

They are both perfect squares and perfect cubes.

This applies specifically to the numbers 1, 64, and 4,096. 729 does not fit completely but confirms that several of the numbers are indeed both perfect squares and cubes. Hence, the overall pattern recognizes these aspects among the listed values.