A perfect square can make a perfect cube when the number of blocks needed to form both the square and cube is the same. This occurs when the side of the square and cube are powers of two. The first time this happens is with 8 blocks forming an 8x8 square and 4x4x4 cube, as 8=23 and 4=22. The next larger numbers that are both perfect squares and cubes are 16, 64, 256, and 1024.

1. WHEN CAN A PERFECT
SQUARE
MAKE A PERFECT CUBE?
Expressions and Equations 8.EE
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of
small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For
example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

2. One square
1 x 1 Square 1x1x1 Cube

3. Four Blocks
2x2 Square not a Cube

4. Nine Blocks
3x3 Square not a Cube

5. 16 Blocks
4x4 Square not a Cube

6. Write
side side 2 side 3
1
2
3
4
5
6
7
8
9

7. Write
side side 2 side 3
1 1x1=1 1x1x1=1
2 2x2=4 2x2x2=8
3 3x3=9 3x3x3=27
4
5
6
7
8
9

8. side side 2 side 3
1 1x1=1 1x1x1=1
2 2x2=4 2x2x2=8
3 3x3=9 3x3x3=27
4 16 64
5 25 125
6 36 216
7 49 343
8 64 512
9 81 729
10 100 1000
11 121 1331
12 144 1728
13 169 2197
14 196 2744

9. 25 Blocks
5x5 Square not a Cube

10. 36 Blocks
6x6 Square not a Cube

11. 49 Blocks
7x7 Square not a Cube

12. 64 Blocks
8x8 Square 4x4x4 Cube

13. Cube = Square
How many pieces will make a cube and
a square?

14. Square = 8x8=64
8=2x2x2
thus 8x8= (2x2x2) x (2x2x2)
2x2x2x2x2x2= 26

15. Cube = 4x4x4=64
4=2x2
On the cube: 4x4x4 can be written:
(2x2) x (2x2) x (2x2) = 26
82 = 43

16. When will it occur again?
1x1x1x1x1x1= 1
2x2x2x2x2x2= 64
????
Find the next 4 numbers that are perfect
squares and perfect cubes.