To solve the problem of finding the quotient of [tex]\(\frac{-8 x^6}{4 x^{-3}}\)[/tex], follow these detailed steps:
1. Separate the constants and the variable terms:
- The constants in the numerator and denominator are [tex]\(-8\)[/tex] and [tex]\(4\)[/tex] respectively.
- The variable terms are [tex]\(x^6\)[/tex] in the numerator and [tex]\(x^{-3}\)[/tex] in the denominator.
2. Simplify the constants:
- Dividing the constants: [tex]\(\frac{-8}{4} = -2\)[/tex].
3. Simplify the variable exponents:
- When dividing like bases, you subtract the exponents of the denominator from the numerator. Here, we have [tex]\(x^6\)[/tex] and [tex]\(x^{-3}\)[/tex].
- So, the exponents calculation is: [tex]\(6 - (-3) = 6 + 3 = 9\)[/tex].
4. Combine the simplified constant and the variable terms:
- The simplified constant term is [tex]\(-2\)[/tex].
- The simplified variable term with the exponent is [tex]\(x^9\)[/tex].
Therefore, the quotient of [tex]\(\frac{-8 x^6}{4 x^{-3}}\)[/tex] is:
[tex]\[
-2 x^9
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{-2 x^9}
\][/tex]
