Doubly Powerful
To what power should $4^4$ be raised to give $8^8$?
Age
14 to 16
Challenge level
Problem
To what power should we raise $4^4$ to get $8^8$?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 3
Using powers of $2$
$4 = 2^2$ and $8 = 2^3$. Therefore:
$4^4 = \left(2^2\right)^4 = 2^{2 \times 4} = 2^8$
$8^8 = \left(2^3\right)^8 = 2^{3 \times 8} = 2^{24}$
This means: $8^8 = 2^{24} = 2^{8 \times 3} = \left(2^8\right)^3 = \left(4^4\right)^3$.
Hence, $4^4$ needs to be cubed to obtain $8^8$.
Using powers of $4$
$\begin{align}8^8 &= \left(2\times4\right)^8\\
&=2^8\times4^8\\
&=2^{2\times4}\times4^8\\
&=\left(2^4\right)^4\times4^8\\
&=4^4\times4^8\\
&=4^4\times4^4\times4^4\\
&=\left(4^4\right)^3\end{align}$