Solve $T(n) = (T(\frac{n}{2}))^3 , T(1) = 2$ Solve $T(n) = (T(\frac{n}{2}))^3 , T(1) = 2$

Well, I'm not sure how to approach this properly. can anyone help me out?

I've tried $T(1) = (T(0.5))^3 = 2$

But that doesn't help me much.

Find: $\Theta$ of the function.
Your equations only define what $T(2^k)$ is equal to for $k\in\mathbb Z$, since

$$\begin{align}T(2)&amp;=T(1)^3=8\\
T(4)&amp;=T(2)^3=8^3=512\\
T(8)&amp;=T(4)^3=512^3\\
\vdots&amp;\end{align}$$

For $x\notin\{2^k| k\in \mathbb Z\}=A$, you cannot (from the given data) calculate $T(x)$.

For example, you can define $T(x)=0$ on all $x\notin A$, and the condition is satisfied, but you can also define $T(x)=1$, and the condition will still be satisfied.
The answer is $\Theta(2^{n ^ {log3}})$