To simplify the expression [tex]\(\left(-\frac{2 x}{y^2}\right)^3\)[/tex], follow these steps:
1. Understand the expression: We're given the expression [tex]\(\left(-\frac{2 x}{y^2}\right)^3\)[/tex]. This means we need to raise [tex]\(-\frac{2 x}{y^2}\)[/tex] to the power of 3.
2. Apply the power to both the numerator and the denominator:
[tex]\[
\left(-\frac{2 x}{y^2}\right)^3 = \left(-2 x\right)^3 \div \left(y^2\right)^3
\][/tex]
3. Simplify the numerator: Raise [tex]\(-2 x\)[/tex] to the power of 3.
- When raising a product to a power, we raise each factor to that power:
[tex]\[
(-2 x)^3 = (-2)^3 \cdot (x)^3
\][/tex]
- Calculate [tex]\((-2)^3\)[/tex]:
[tex]\[
(-2)^3 = -2 \cdot -2 \cdot -2 = -8
\][/tex]
- Raise [tex]\(x\)[/tex] to the power of 3:
[tex]\[
x^3 = x^3
\][/tex]
- Therefore:
[tex]\[
(-2 x)^3 = -8 x^3
\][/tex]
4. Simplify the denominator: Raise [tex]\(y^2\)[/tex] to the power of 3.
[tex]\[
(y^2)^3 = y^{2 \cdot 3} = y^6
\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[
\left(-\frac{2 x}{y^2}\right)^3 = \frac{-8 x^3}{y^6}
\][/tex]
Thus, the simplified form of the expression [tex]\(\left(-\frac{2 x}{y^2}\right)^3\)[/tex] is:
[tex]\[
\boxed{-\frac{8 x^3}{y^6}}
\][/tex]
