To solve the equation \( z^2 = 109.434613210000 \), you need to take the square root of both sides.
So,
\[ z = \sqrt{109.434613210000} \]
Calculating the square root gives:
\[ z \approx 10.4626 \]
Rounding this to the nearest tenth, we get:
\[ z \approx 10.5 \]
Thus, the final answer is \( z \approx 10.5 \).