To find the value of [tex]\( x^{-11} - 2017 \)[/tex], where [tex]\( x = \frac{20172016^2}{20172015^2 + 20172017^2 - 2} \)[/tex], we will follow the following steps:
1. Compute [tex]\( x \)[/tex]:
[tex]\[
x = \frac{(20172016)^2}{(20172015)^2 + (20172017)^2 - 2}
\][/tex]
2. Evaluate the exponentiation [tex]\( x^{-11} \)[/tex]:
[tex]\[
x^{-11} = \left( \frac{20172016^2}{20172015^2 + 20172017^2 - 2} \right)^{-11}
\][/tex]
3. Subtract 2017 from the result of [tex]\( x^{-11} \)[/tex]:
[tex]\[
x^{-11} - 2017
\][/tex]
Performing these steps yields:
[tex]\[
x^{-11} - 2017 = 31
\][/tex]
Thus, the value of [tex]\( x^{-11} - 2017 \)[/tex] is [tex]\(\boxed{31}\)[/tex].
