The problem requires simplifying two terms, each raised to the eighth power. The expressions within the parentheses are structured to represent hyperbolic cosine functions.Step 1: Determine the simplification approach.
\[{Since } (x+\sqrt{x^2-1})^8 + (x-\sqrt{x^2-1})^8\]
\[= 2\{{}^8C_0 x^8 + {}^8C_2 x^6 (x^2-1) + {}^8C_4 x^4 (x^2-1)^2 + {}^8C_6 x^2 (x^2-1)^3 + {}^8C_8 x^0 (x^2-1)^4 \}\]
\[{The coefficient of the highest power of } x \]
\[= 2\{{}^8C_0 + {}^8C_2 + {}^8C_4 + {}^8C_6 + {}^8C_8 \}\]
\[= (1+1)^8 + (1-1)^8 = 2^8 = 256\] This value represents the highest power and aligns with option (C).