Can you solve it with speed equals to c?

$\begin{aligned} \sqrt{117^2-108^2} & = \sqrt{(117+108)(117-108)} \\ & = \sqrt{(225)(9)} \\ & = \sqrt{15^2\times 3^2} \\ & = 15\times 3 \\ & = \boxed{45} \end{aligned}$

Quick check using sum of digits shows that both $117$ and $108$ are divisible by $9$ . That gives us

$\sqrt{117^{2}-108^{2}} = 9\sqrt{13^{2}-12^{2}}$

We know that the triple $5-12-13$ forms a right triangle; so, by Pythagoras's theorem, we get that

$9\sqrt{13^{2}-12^{2}} = 9\sqrt{5^{2}} = 9 \times 5 = \boxed{45}$

Nope. You are overestimating the power of a brain or underestimating relativity

$\sqrt{225\times 9}$ using identity of $a^2-b^2$

$15\times 3=45$