## Double perfect squares

We have recently done a unit on perfect squares in school. The following conversation with Katie (8yo) happened completely spontaneously on a peaceful, evening ride home.

K: 10,000 is a double perfect square.
Me: What do you mean?
K: Well, 100 is already a perfect square, and 10,000 is 100 times 100.
Me: I see. Can you think of another double perfect square.
K: Yeah, 81.
Me: Indeed. Now, double perfect squares are also called perfect fourth powers.
K: What does that mean?
Me: It means that you can get them by multiplying a number by itself 4 times.
K: I don’t really understand.
Me: Well, 100 is 10 times 10 so 100 times 100 is 10 times 10 times 10 times 10, that is, 10 times itself 4 times. So what is 81 the fourth power of?
K: 3, because it is 3 times 3 times 3 times 3.
Me: Yep. Now if a number can be obtained by multiplying some number by itself 3 times, then it is called a perfect cube.
K: How come?
Me: Well it has something to do with cubes. Why don’t you think about it.

I am totally going to use the term “double perfect square” now!

To be continued…