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To solve the given expression, let's break it down step by step.

The expression inside the square root is:

\[
\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}} \times x^{\frac{8}{3}}}
\]

### Step 1: Simplify the denominator
In the denominator, you have \( x^{\frac{1}{3}} \times x^{\frac{8}{3}} \). When multiplying terms with the same base, you add their exponents:

\[
x^{\frac{1}{3}} \times x^{\frac{8}{3}} = x^{\frac{1}{3} + \frac{8}{3}} = x^{\frac{9}{3}} = x^3
\]

### Step 2: Simplify the entire expression inside the square root
Now the expression becomes:

\[
\frac{x^{\frac{4}{3}}}{x^3}
\]

When you divide terms with the same base, you subtract the exponents:

\[
\frac{x^{\frac{4}{3}}}{x^3} = x^{\frac{4}{3} - 3} = x^{\frac{4}{3} - \frac{9}{3}} = x^{\frac{-5}{3}}
\]

### Step 3: Apply the square root
The square root of \( x^{\frac{-5}{3}} \) is:

\[
\sqrt{x^{\frac{-5}{3}}} = x^{\frac{-5}{6}}
\]

### Final Answer
The simplified expression is:

\[
\boxed{x^{\frac{-5}{6}}}
\]

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

### Related Questions:
1. How do you simplify expressions involving square roots and exponents?
2. What are the rules for multiplying and dividing exponents with the same base?
3. How do you convert negative exponents into positive ones?
4. How do you simplify expressions with fractional exponents?
5. How do you interpret the result of taking a square root of a term with a fractional exponent?

### Tip:
When working with fractional exponents, remember that the numerator of the exponent indicates the power, and the denominator indicates the root. For example, \( x^{\frac{m}{n}} \) means the \( n \)-th root of \( x^m \).

How do you solve the expression \( \sqrt{\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}} \times x^{\frac{8}{3}}}} \)?

Learn how to simplify the radical expression \( \sqrt{\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}} \times x^{\frac{8}{3}}}} \) with a detailed step-by-step solution. This problem covers key algebraic concepts including exponent rules, product and quotient of powers, and the square root of powers. Suitable for students in Grades 10-12.

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