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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 \(\frac{a^m}{a^n} = a^{m-n}\).

Learn how to simplify the expression (xy^2)^{-6} * (x^3 y^7)^2 / (x^0 y^2)^{-0.5} using exponent rules. This problem covers key concepts in exponents and fractional exponents. Suitable for Grades 9-12.

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