Consider $$a^b= a ^ {101101} $$
As if we split the binary representation of $b$,
$$b = 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2+ 0 \cdot 2^1 + 1 \cdot 2^0 $$
Then how are we able to write $$a^b = ( 1 \cdot (a ^{2^5} ) \cdot ( 0 \cdot(a ^{2^4} ) ) \cdot( 1 \cdot (a ^{2^3} ) ) \cdot ( 1 \cdot (a ^{2^2} ) ) \cdot ( 0 \cdot (a ^{2^1} ) ) \cdot ( 1 \cdot (a ^{2^0} ) ) $$
$$\begin{align*} a^{101101_2} &= a^{1\cdot2^5+0\cdot2^4+1\cdot2^3+1\cdot2^2 + 0\cdot2^1+1\cdot 2^0}\\ &= a^{2^5} \cdot\left(a^{2^4}\right)^0 \cdot a^{2^3} \cdot a^{2^2}\cdot\left(a^{2^1}\right)^0\cdot a^{2^0}\\ &= a^{2^5} \cdot\left(a^0\right)^{2^4} \cdot a^{2^3} \cdot a^{2^2}\cdot\left(a^0\right)^{2^1}\cdot a^{2^0}\\ &= a^{2^5} \cdot 1 \cdot a^{2^3} \cdot a^{2^2}\cdot 1\cdot a^{2^0}\\ &= a^{2^5} \cdot a^{2^3} \cdot a^{2^2}\cdot a^{2^0}\\ \end{align*}$$