To find the expression equivalent to [tex]\(\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}\)[/tex], let's simplify it step by step.
1. Separate the coefficients and the variables:
[tex]\[
\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}} = \frac{10}{-5} \cdot \frac{x^6}{x^{-2}} \cdot \frac{y^{12}}{y^{-6}}
\][/tex]
2. Simplify the coefficient part:
[tex]\[
\frac{10}{-5} = -2
\][/tex]
3. Simplify the [tex]\(x\)[/tex] part using the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8
\][/tex]
4. Simplify the [tex]\(y\)[/tex] part using the same exponent rule:
[tex]\[
\frac{y^{12}}{y^{-6}} = y^{12 - (-6)} = y^{12 + 6} = y^{18}
\][/tex]
Putting everything together, we get:
[tex]\[
-2 \cdot x^8 \cdot y^{18}
\][/tex]
Hence, the expression equivalent to [tex]\(\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}\)[/tex] is:
[tex]\[
-2 x^8 y^{18}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{-2 x^8 y^{18}}
\][/tex]
