One thing you may have noticed about the previous diagram is that the bottom row clusters contain the square of the number of dots in the cluster directly above. Thus, we could label the diagram as:

However, the bottom row clusters have 1, 4, 16, and 64 dots sequentially. That is, in base 2 exponentials, 2^0, 2^2, 2^4, and 2^6. Therefore, we see:

\begin{align}\left(2^0\right)^2= & 2^0\\ \left(2^1\right)^2= & 2^2\\ \left(2^2\right)^2= & 2^4\\ \left(2^3\right)^2= & 2^6.\end{align}

You may notice the pattern that the exponential of an exponential simplifies to a single exponential to the power of the product of the original powers.

Here is another example highlighting the algebra behind this pattern:

\left(3^2\right)^3=\left(3\cdot3\right)\cdot\left(3\cdot3\right)\cdot\left(3\cdot3\right)=3\cdot3\cdot3\cdot3\cdot3\cdot3=3^6.

Exponential Algebra