To solve the expression [tex]\(\left(\frac{-2 x^2 y^3}{x^3 y}\right)^4\)[/tex], we'll follow a detailed step-by-step approach.
1. Simplify the fraction inside the parentheses:
The given expression is:
[tex]\[
\left(\frac{-2 x^2 y^3}{x^3 y}\right)^4
\][/tex]
2. Simplify the numerator and the denominator separately:
- The numerator is [tex]\(-2 x^2 y^3\)[/tex].
- The denominator is [tex]\(x^3 y\)[/tex].
3. Divide the terms in the numerator by the corresponding terms in the denominator:
[tex]\[
\frac{-2 x^2 y^3}{x^3 y}
\][/tex]
We can break this down as follows:
- The constant term stays the same: [tex]\(-2\)[/tex].
- For the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}
\][/tex]
- For the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^3}{y} = y^{3-1} = y^2
\][/tex]
Putting this all together, the simplified fraction becomes:
[tex]\[
\frac{-2 x^2 y^3}{x^3 y} = -2 \cdot \frac{1}{x} \cdot y^2 = -\frac{2y^2}{x}
\][/tex]
4. Raise the simplified fraction to the power of 4:
Now we need to raise [tex]\(\left(-\frac{2y^2}{x}\right)\)[/tex] to the power of 4:
[tex]\[
\left(-\frac{2y^2}{x}\right)^4
\][/tex]
Apply the power to each component inside the parentheses:
[tex]\[
\left(-\frac{2y^2}{x}\right)^4 = \left(-1\right)^4 \cdot \left(2y^2\right)^4 \cdot \left(\frac{1}{x}\right)^4
\][/tex]
Simplify each part:
- [tex]\((-1)^4 = 1\)[/tex]
- [tex]\((2y^2)^4 = 2^4 \cdot (y^2)^4 = 16 \cdot y^{8}\)[/tex]
- [tex]\(\left(\frac{1}{x}\right)^4 = \frac{1}{x^4}\)[/tex]
Putting it all together, we get:
[tex]\[
1 \cdot 16 y^8 \cdot \frac{1}{x^4} = \frac{16 y^8}{x^4}
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\left(\frac{-2 x^2 y^3}{x^3 y}\right)^4\)[/tex] is:
[tex]\[
\boxed{\frac{16 y^8}{x^4}}
\][/tex]
