Algebraic Manipulation (Problem 1)

Conditions important:

• Power of $4$ is only important because it includes the power of $2$ and fractional powers of $16$
• Digit sum is a prime $\leq 10$

Integers that satisfy all three conditions are all powers of four: $4, 16, 64, 256, 1024 \ldots$ . Now, we can calculate whether x is an integer for these values

$\begin{aligned} \sqrt{(x+12)^{5}} = 4 &\implies x \ne \Z \\ \sqrt{(x+12)^{5}} = 16 &\implies x \ne \Z \\ \sqrt{(x+12)^{5}} = 64 &\implies x \ne \Z \\ \sqrt{(x+12)^{5}} = 256 &\implies x \ne \Z \\ \color{#20A900}{\sqrt{(x+12)^{5}} = 1024} & \color{#20A900} \implies x = 4 \end{aligned}$

Since we are asked the value of $\sqrt{(x+12)^{5}}$ for $x = 4$ . We get the answer as $\boxed{\color{#20A900}{1024}}$