an exponential equation that goes through (3,64) and (5,1024)

an exponential equation that goes through (3,64) and (5,1024)

$$\small{\text{ Exponential equation is: $\quad \boxed{y=y_0*a^x}$ }}\\ \small{\text{ $ (1): \quad 64 = y_0*a^3 \quad | \quad (3,64) $ }}\\ \small{\text{ $ (2): \quad 1024 = y_0*a^5 \quad | \quad (5,1024) $ }}\\\\ \begin{array}{lrcl} \hline \\ \dfrac{(2)}{(1)}: &\dfrac{1024}{64} &=& \dfrac{y_0a^5}{y_0a^3} \\\\ &\dfrac{1024}{64} &=& \dfrac{a^5}{a^3} \\ \\ &16 &=&a^{5-3} \\ &16 &=&a^{2} \quad | \quad \sqrt\\ &4 &=&a\\ &a &=&4 \end{array} \begin{array}{lrcl} \hline \\ (1): & 64 &=& y_0*4^3 \\ & 64 &=& y_0*64 \\ & 1 &=& y_0\\ & y_0 &=& 1\\ \end{array} $\quad \small{\text{The exponential equation is: }}\boxed{y=1*4^x=4^x}$$ $

$$\\ \small{\text{ Proof: }}\\ \small{\text{ $ \quad y(3) = 4^3 = 64 \quad okay! $ }}\\ \small{\text{ $ \quad y(5) = 5^3 = 1024 \quad okay! $ }}$$