The given expression is:

$\sqrt[3]{\frac{x^6}{12b}}$

To convert this radical expression to exponential notation:

1. Recall that a cube root $\sqrt[3]{a}$ can be written as $a^{\frac{1}{3}}$.
2. The fraction $\frac{x^6}{12b}$ can be kept as is for now.
3. Applying the cube root to both the numerator and the denominator, we have:

\left(\frac{x^6}{12b}\right)^{\frac{1}{3}} = \frac{x^6^{\frac{1}{3}}}{(12b)^{\frac{1}{3}}}

4. Simplify x^6^{\frac{1}{3}} as $x^2$, since $(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2$.

Thus, the expression becomes:

$\frac{x^2}{(12b)^{\frac{1}{3}}}$

This is the expression converted to exponential notation.

Let me know if you'd like further details or have any questions. Here are some related questions:

1. How do you simplify other fractional exponents?
2. What is the general rule for converting $n$-th roots to exponents?
3. How do you handle roots when variables have negative exponents?
4. Can you combine radicals with different indices using exponential notation?
5. How does simplifying fractions affect exponential notation?

Tip: Always remember that the $n$-th root of a number can be rewritten as raising the number to the power of $\frac{1}{n}$.