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To solve this problem, let's go through each part of the expression step-by-step. We need to simplify the expression on the left side so it matches the form \( x^a y^b \).

Given:

\[
\frac{\sqrt{x^{-1} \cdot x^{\frac{3}{2}} y^2}}{\sqrt{x^6 y^{-\frac{2}{3}}}} = x^a y^b
\]

### Step 1: Simplify the Numerator
In the numerator, we have:
\[
\sqrt{x^{-1} \cdot x^{\frac{3}{2}} y^2}
\]
This can be simplified by combining the powers of \( x \) and \( y \) within the square root.

1. **Combine powers of \( x \):**
   \[
   x^{-1} \cdot x^{\frac{3}{2}} = x^{-1 + \frac{3}{2}} = x^{\frac{1}{2}}
   \]

2. **Simplify \( y^2 \):** Since \( y^2 \) is already a single term, we keep it as is.

Now the expression inside the square root becomes:
\[
x^{\frac{1}{2}} y^2
\]

Taking the square root, we get:
\[
\sqrt{x^{\frac{1}{2}} y^2} = x^{\frac{1}{4}} y
\]

### Step 2: Simplify the Denominator
In the denominator, we have:
\[
\sqrt{x^6 y^{-\frac{2}{3}}}
\]
Taking the square root of each term individually:

1. For \( x^6 \):
   \[
   \sqrt{x^6} = x^3
   \]

2. For \( y^{-\frac{2}{3}} \):
   \[
   \sqrt{y^{-\frac{2}{3}}} = y^{-\frac{1}{3}}
   \]

So the denominator simplifies to:
\[
x^3 y^{-\frac{1}{3}}
\]

### Step 3: Combine Numerator and Denominator
Now, the expression becomes:
\[
\frac{x^{\frac{1}{4}} y}{x^3 y^{-\frac{1}{3}}}
\]

Separate the terms for \( x \) and \( y \):

1. For \( x \):
   \[
   \frac{x^{\frac{1}{4}}}{x^3} = x^{\frac{1}{4} - 3} = x^{\frac{1}{4} - \frac{12}{4}} = x^{-\frac{11}{4}}
   \]

2. For \( y \):
   \[
   \frac{y}{y^{-\frac{1}{3}}} = y^{1 + \frac{1}{3}} = y^{\frac{4}{3}}
   \]

Thus, we have:
\[
x^{-\frac{11}{4}} y^{\frac{4}{3}} = x^a y^b
\]

### Step 4: Identify \( a \) and \( b \)
By comparison, we find:
\[
a = -\frac{11}{4} \quad \text{and} \quad b = \frac{4}{3}
\]

### Final Answer
\[
a = -\frac{11}{4}, \quad b = \frac{4}{3}
\]

Would you like further clarification on any step? 

---

### Related Questions

1. How do you simplify expressions with fractional exponents?
2. What are the rules for combining powers with the same base?
3. How can square roots be expressed in terms of exponents?
4. How do you handle negative exponents in fractions?
5. What is the process for rationalizing expressions with exponents?

### Tip
When simplifying expressions with multiple terms under a square root, it can help to split up the terms and simplify each part separately.

Given that \( \frac{\sqrt{x^{-1} \cdot x^{\frac{3}{2}} y^2}}{\sqrt{x^6 y^{-\frac{2}{3}}}} = x^a y^b \), find \( a \) and \( b \).

Learn how to find the values of \( a \) and \( b \) in the expression \( \frac{\sqrt{x^{-1} \cdot x^{\frac{3}{2}} y^2}}{\sqrt{x^6 y^{-\frac{2}{3}}}} = x^a y^b \). This problem involves simplifying terms with exponents and square roots, applying exponent rules, and understanding properties of radicals. Suitable for high school students in Grades 10-12.

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