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 $x^m$.