Which power?
Which power of $16$ is equal to $64^6$?
Which power of $16$ is equal to $64^6$?
This problem is adapted from the UKMT Mathematical Challenges.
Answer: $9$
Writing it out
$64^6=64\times64\times64\times64\times64\times64$, and $64=16\times4$. So $$\begin{split}64^6&=(16\times4)\times(16\times4)\times(16\times4)\times(16\times4)\times(16\times4)\times(16\times4)\\
&=16\times16\times16\times16\times16\times16\times4\times4\times4\times4\times4\times4\\
&=16\times16\times16\times16\times16\times16\times \hspace{2.9mm}16\hspace{2.9mm} \times \hspace{2.9mm}16\hspace{2.9mm} \times\hspace{2.9mm}16\\
&=16^9\end{split}$$
Using index laws
$64=16\times4$, so $64^6=(16\times4)^6$
$=16^6\times4^{2\times3}$
$=16^6\times\left(4^2\right)^3$
But $4^2=16$, so $16^6\times\left(4^2\right)^3=16^6\times16^3=16^9$
Writing both numbers in terms of 4
$16=4^2$ and $64=4^3$
So $64^6=\left(4^3\right)^6=4^{3\times6}$
So $64^6=4^{18}=4^{2\times9}=\left(4^2\right)^9=16^9$