The given expression is:

$\sqrt[3]{\sqrt{x^7}}$

To simplify it, follow these steps:

Step 1: Rewrite the expression using rational exponents.

We know that: $\sqrt{x^7} = x^{7/2}$ Thus, the original expression becomes: $\sqrt[3]{x^{7/2}} = x^{(7/2) \cdot (1/3)}$

Step 2: Simplify the exponent.

$\left(\frac{7}{2} \cdot \frac{1}{3}\right) = \frac{7}{6}$

Step 3: Final expression.

The simplified form is: $x^{7/6}$

So, the simplest radical form of the given expression is $x^{7/6}$.

Would you like more details on any part of the process?

Here are five related questions:

1. How do you simplify expressions with fractional exponents?
2. What is the relationship between roots and exponents?
3. How do you simplify higher-order roots of powers?
4. Can you explain the rules of multiplying exponents?
5. What are some real-world applications of radical expressions?

Tip: When simplifying radicals, remember that square roots and cube roots can often be expressed as fractional exponents for easier manipulation.