To find the quotient for the expression [tex]\(\frac{-8 x^6}{4 x^{-3}}\)[/tex], let's break it down step-by-step.
1. Simplify the Constants:
We have [tex]\(\frac{-8}{4}\)[/tex] from the constants in the numerator and denominator.
[tex]\[
\frac{-8}{4} = -2
\][/tex]
2. Simplify the Exponents on [tex]\( x \)[/tex]:
We need to divide [tex]\( x^6 \)[/tex] by [tex]\( x^{-3} \)[/tex]. We do this by subtracting the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^6 \div x^{-3} = x^{6 - (-3)} = x^{6 + 3} = x^9
\][/tex]
3. Combine the Simplified Parts:
Now, we combine the simplified constant part with the simplified exponent part:
[tex]\[
-2 \times x^9 = -2 x^9
\][/tex]
Therefore, the quotient of [tex]\(\frac{-8 x^6}{4 x^{-3}}\)[/tex] is:
[tex]\[
-2 x^9
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{-2 x^9}
\][/tex]
