To simplify the expression $\left( \frac{x^6}{27} \right)^{\frac{1}{3}}$, we can use the properties of exponents and radicals.

Solution:

1. Start with the given expression: $\left( \frac{x^6}{27} \right)^{\frac{1}{3}}$

2. Apply the exponent $\frac{1}{3}$ to both the numerator and the denominator separately: $\frac{(x^6)^{\frac{1}{3}}}{27^{\frac{1}{3}}}$

3. Simplify each part:

   • For the numerator $(x^6)^{\frac{1}{3}}$: $(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2$
   • For the denominator $27^{\frac{1}{3}}$: 27^{\frac{1}{3}} = 3 \quad \text{(since 27 = 3^3)}

4. Substitute back: $\frac{x^2}{3}$

Final Answer:

$\frac{x^2}{3}$

Would you like further details or have any questions?

Here are some related questions you might find useful:

1. How do you simplify expressions with fractional exponents?
2. What is the process for taking roots of fractions?
3. How can we apply the rules of exponents to simplify radical expressions?
4. What if the exponent were $\frac{1}{2}$ instead of $\frac{1}{3}$?
5. How do we handle negative exponents in similar expressions?

Tip: Remember that fractional exponents represent roots; for example, $x^{\frac{1}{3}}$ is the cube root of $x$.