$$ \sqrt{x} $$$$ \sqrt{x^2y^2}\sqrt{xy^2} $$

Let's break down and simplify each of these expressions:

1. $\sqrt{x}$:
   - This is the square root of $x$. It remains as is, unless you specify any further conditions on $x$.

2. $\sqrt{x^2 y^2} \sqrt{xy^2}$:
   - First, simplify $\sqrt{x^2 y^2}$:
     \[
     \sqrt{x^2 y^2} = \sqrt{x^2} \cdot \sqrt{y^2} = x \cdot y
     \]
   - Now simplify $\sqrt{xy^2}$:
     \[
     \sqrt{xy^2} = \sqrt{x} \cdot \sqrt{y^2} = \sqrt{x} \cdot y
     \]
   - Combining both parts:
     \[
     \sqrt{x^2 y^2} \cdot \sqrt{xy^2} = (x \cdot y) \cdot (\sqrt{x} \cdot y) = x \cdot y^2 \cdot \sqrt{x}
     \]
   - So the simplified expression is:
     \[
     x y^2 \sqrt{x}
     \]

### Final answer:
\[
\sqrt{x} \quad \text{and} \quad x y^2 \sqrt{x}
\]

Would you like further clarification or details?

### Related Questions:
1. How would this expression change if $x$ or $y$ were negative?
2. What happens when you take the cube root instead of the square root?
3. Can you simplify expressions involving cube roots in a similar manner?
4. How does the simplification change when fractional exponents are used instead of radicals?
5. How would this expression appear if written using exponents rather than square roots?

### Tip:
Always break down complex square roots into smaller components to simplify them step by step.

$$ \sqrt{x} $$ $$ \sqrt{x^2y^2}\sqrt{xy^2} $$

Learn how to simplify the radical expressions $$ \sqrt{x} $$ and $$ \sqrt{x^2 y^2}\sqrt{xy^2} $$ with step-by-step guidance. This problem focuses on algebraic concepts such as square roots and simplifying radicals, suitable for students in Grades 9-10.

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