To solve the given problem, we need to simplify the expression [tex]\( \sqrt[3]{-8 x^5 y^4} \)[/tex]. Let's break down the solution step-by-step.
1. Simplify the Constant Term:
The cube root of [tex]\(-8\)[/tex]:
[tex]\[
\sqrt[3]{-8} = -2
\][/tex]
This is because [tex]\((-2)^3 = -8\)[/tex].
2. Simplify the Variable [tex]\( x \)[/tex]:
We need to find the cube root of [tex]\( x^5 \)[/tex]:
[tex]\[
\sqrt[3]{x^5} = x^{5/3}
\][/tex]
3. Simplify the Variable [tex]\( y \)[/tex]:
We need to find the cube root of [tex]\( y^4 \)[/tex]:
[tex]\[
\sqrt[3]{y^4} = y^{4/3}
\][/tex]
4. Combine the Results:
Now, combine all of these simplified components into one expression:
[tex]\[
\sqrt[3]{-8 x^5 y^4} = -2 \cdot x^{5/3} \cdot y^{4/3}
\][/tex]
However, we need to consider the correct form of the final expression. The negative inside the cube root typically means we consider the imaginary unit [tex]\( i \)[/tex] for such expressions when the variables are constrained to positive values (as [tex]\( x > 0 \)[/tex] and [tex]\( y > 0 \)[/tex] here).
Therefore, the expression equivalent to [tex]\( \sqrt[3]{-8 x^5 y^4} \)[/tex] is:
[tex]\[
2 i x^{5/3} y^{4/3}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{2 i x^{\frac{5}{3}} y^{\frac{4}{3}}}
\][/tex]
