To simplify the given expression [tex]\(\frac{x^{-8}}{-11 x^7}\)[/tex], follow these steps:
1. Rewrite the expression by using the properties of exponents: Recall that [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex] and [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]. The original expression can be rearranged as follows:
[tex]\[
\frac{x^{-8}}{-11 x^7} = \frac{x^{-8}}{(-11) \cdot x^7}
\][/tex]
2. Combine the exponents: Since exponents with the same base in the numerator and denominator can be subtracted, we have:
[tex]\[
x^{-8 - 7} = x^{-15}
\][/tex]
This simplifies our expression to:
[tex]\[
\frac{1}{-11 x^{15}}
\][/tex]
3. Simplify the overall expression: Observing the sign and coefficient, we recognize that the negative sign can simply be taken to the front of the fraction:
[tex]\[
-\frac{1}{11 x^{15}}
\][/tex]
Hence, the simplified form of the given expression [tex]\(\frac{x^{-8}}{-11 x^7}\)[/tex] is:
[tex]\[
-\frac{1}{11 x^{15}}
\][/tex]
So, the correct answer from the given options is:
[tex]\(\boxed{-\frac{1}{11 x^{15}}}\)[/tex].
