The problem asks you to compare two numbers raised to the power of 3: $\left( \frac{1}{3} \right)^3$ and $\left( \frac{1}{4} \right)^3$, and then select the appropriate symbol $<$, $=$, or $>$ for the comparison.

Let's break it down:

1. Evaluate $\left( \frac{1}{3} \right)^3$: $\left( \frac{1}{3} \right)^3 = \frac{1}{3^3} = \frac{1}{27}$

2. Evaluate $\left( \frac{1}{4} \right)^3$: $\left( \frac{1}{4} \right)^3 = \frac{1}{4^3} = \frac{1}{64}$

Now compare the two fractions $\frac{1}{27}$ and $\frac{1}{64}$. Since $\frac{1}{27}$ is greater than $\frac{1}{64}$ (the smaller the denominator, the larger the fraction), we have: $\frac{1}{27} > \frac{1}{64}$

Thus, the correct answer is $>$.

Let me know if you need further details or clarifications! Here are five questions to expand your understanding:

1. What happens when the base of a fraction is negative and raised to an odd power?
2. How would the comparison change if the exponents were different?
3. How do you compare powers of fractions with different bases?
4. What general rule can be used to compare fractions raised to powers?
5. How would the result change if the exponents were negative?

Tip: When comparing fractions raised to a power, the base of the fraction plays a significant role; smaller fractions become even smaller when raised to higher powers.