How is this worked out manually?

That is the calculator method.

How to work out 8^(5/3) manually.

$8^{\frac{5}{3}}=(\sqrt[3]{8})^5=2^5=2\cdot 2 \cdot 2 \cdot 2 \cdot 2 = 4 \cdot 2 \cdot 2 \cdot 2 = 8 \cdot 2 \cdot 2 = 16 \cdot 2 = 32\\ \text{Or if you know what's 2^5, the answer is straight-foward-ly 32.}$

See your latest post. Melody replied to your post about the derivative one.