To solve the given expression:

$\frac{(xy^2)^{-6} \times (x^3 y^7)^2}{(x^0 y^2)^{-0.5}}$

we will simplify each component step by step.

Step 1: Simplify each term

1. $(xy^2)^{-6}$: $(xy^2)^{-6} = x^{-6} \cdot (y^2)^{-6} = x^{-6} \cdot y^{-12}$

2. $(x^3 y^7)^2$: $(x^3 y^7)^2 = x^{3 \cdot 2} \cdot y^{7 \cdot 2} = x^6 \cdot y^{14}$

3. $(x^0 y^2)^{-0.5}$: $(x^0 y^2)^{-0.5} = x^{0 \cdot -0.5} \cdot y^{2 \cdot -0.5} = x^0 \cdot y^{-1} = y^{-1}$

Step 2: Combine the terms

Now substitute back into the original expression: $\frac{x^{-6} y^{-12} \cdot x^6 y^{14}}{y^{-1}}$

Step 3: Simplify the numerator

Combine the exponents in the numerator: $x^{-6 + 6} \cdot y^{-12 + 14} = x^0 \cdot y^2 = y^2$

Step 4: Simplify the division

$\frac{y^2}{y^{-1}}$

Using the properties of exponents: $y^{2 - (-1)} = y^{2 + 1} = y^3$

So, the simplified result is: $\boxed{y^3}$

Would you like more details on any of the steps or have any questions?

Here are some related questions for further practice:

1. Simplify $(x^2 y^3)^{-4}$.
2. Simplify $(a^5 b^{-2})^3 \cdot (a^{-1} b^4)^2$.
3. Simplify $\frac{(p^3 q^2)^{-2}}{(p q^4)^-1}$.
4. Simplify $\frac{(x^{-1} y^3)^2}{(x^2 y^{-2})^{-1}}$.
5. Simplify $(a^0 b^2)^{-3} \cdot (a^4 b^{-5})$.
6. Simplify $(m^3 n^2)^{-4} \cdot (m^2 n^{-3})^2$.
7. Simplify $\frac{(c^4 d^{-3})^2}{(c^{-2} d^2)^3}$.
8. Simplify $\frac{(x^2 y^5)^3}{(x^4 y^{-2})^2}$.

Tip: When dealing with exponents, remember the power rule $(a^m)^n = a^{m \cdot n}$, the product rule $a^m \cdot a^n = a^{m+n}$, and the quotient rule $\frac{a^m}{a^n} = a^{m-n}$.