How to make clear sense of this re-write of an equation

Question

I'm having a little trouble intuitively seeing the step being performed here.

$2\times(2\times(2\times(2\times(2\times3+3)+3)+3)+3)+3) = 2^5\times3 + 2^4\times3 + 2^3\times3 + 2^2\times3 + 2^1\times3 + 2^0\times3$

Answer 1

Perhaps start with the innermost bracket. It should be clear where $2^5\times 3$ comes from (all the terms from the left are multiples of 2). The other term in this bracket will multiply by the same thing, except one less power of 2. From there, you can hopefully see the pattern.

Answer 2

From the inside out:

$$\begin{align*} 2\cdot3+3&=2^1\cdot3+2^0\cdot3\\ 2(2\cdot3+3)+3&=2(2^1\cdot3+2^0\cdot3)+3\\ &=2^2\cdot3+2^1\cdot3+2^0\cdot3\\ 2(2(2\cdot3+3)+3)+3&=2(2^2\cdot3+2^1\cdot3+2^0\cdot3)+3\\ &=2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3\\ 2(2(2(2\cdot3+3)+3)+3)+3&=2(2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3)\\ &=2^4\cdot3+2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3\\ 2(2(2(2(2\cdot3+3)+3)+3)+3)+3&=2(2^4\cdot3+2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3)+3\\ &=2^5+2^4\cdot3+2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3 \end{align*}$$

From the outside in:

$$\begin{align*} 2\color{green}(2\color{red}(2\color{blue}(2\color{brown}(2\cdot3+\color{brown}{3)}+\color{blue}{3)}+\color{red}{3)}+\color{green}{3)}+\color{purple}3&=2^2\color{red}(2\color{blue}(2\color{brown}(2\cdot3+\color{brown}{3)}+\color{blue}{3)}+\color{red}{3)}+2\cdot\color{green}3+\color{purple}3\\ &=2^3\color{blue}(2\color{brown}(2\cdot3+\color{brown}{3)}+\color{blue}{3)}+2^2\cdot\color{red}3+2\cdot\color{green}3+\color{purple}3\\ &=2^4\color{brown}(2\cdot3+\color{brown}{3)}+2^3\cdot\color{blue}3+2^2\cdot\color{red}3+2\cdot\color{green}3+\color{purple}3\\ &=2^5\cdot3+2^4\cdot\color{brown}3+2^3\cdot\color{blue}3+2^2\cdot\color{red}3+2\cdot\color{green}3+\color{purple}3\\ &=2^5\cdot3+2^4\cdot3+2^3\cdot3+2^2\cdot3+2^1\cdot3+2^0\cdot3\\ \end{align*}$$