To simplify the expression [tex]\(\frac{2 x^{-4}}{3 x y}\)[/tex], let's proceed through the following steps:
1. Rewrite the expression using properties of exponents:
The given expression is:
[tex]\[
\frac{2 x^{-4}}{3 x y}
\][/tex]
2. Combine the [tex]\(x\)[/tex] terms in the denominator and numerator:
We have [tex]\(x^{-4}\)[/tex] in the numerator and [tex]\(x\)[/tex] (which is [tex]\(x^1\)[/tex]) in the denominator. According to the properties of exponents, when dividing like bases, we subtract the exponents:
[tex]\[
\frac{x^{-4}}{x} = x^{-4 - 1} = x^{-5}
\][/tex]
3. Substitute back into the original expression:
Now the expression becomes:
[tex]\[
\frac{2 x^{-5}}{3 y}
\][/tex]
4. Simplify the expression further:
An exponent of [tex]\(-5\)[/tex] indicates the reciprocal with a positive exponent:
[tex]\[
x^{-5} = \frac{1}{x^5}
\][/tex]
5. Combine this back into the fraction:
[tex]\[
\frac{2 x^{-5}}{3 y} = \frac{2 \cdot \frac{1}{x^5}}{3 y} = \frac{2}{3 x^5 y}
\][/tex]
So the simplified form of the given expression is:
[tex]\[
\frac{2}{3 x^5 y}
\][/tex]
Now, let’s compare this with the given choices:
A. [tex]\(\frac{4}{x^3 y^6}\)[/tex] - This is not equivalent.
B. [tex]\(4 y^2\)[/tex] - This is not equivalent.
C. [tex]\(\frac{4 x^3 y^2}{3}\)[/tex] - This is not equivalent.
D. [tex]\(\frac{2}{3 x^5 y}\)[/tex] - This matches our simplified result.
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
