Let's break down and solve the expression $\frac{(\sqrt{7})^3}{7}$.

1. First, simplify $(\sqrt{7})^3$. The square root of 7 raised to the third power is: $(\sqrt{7})^3 = 7^{3/2} = 7 \times \sqrt{7}$

2. Now, divide this by 7: $\frac{7 \times \sqrt{7}}{7}$

3. The 7's cancel out, leaving: $\sqrt{7}$

Thus, the simplified result is $\sqrt{7}$.

Would you like further clarification or more details?

Here are 5 related questions to expand your understanding:

1. How do you simplify expressions with square roots and exponents?
2. What are the properties of exponents when dealing with square roots?
3. How do you simplify expressions like $\frac{a^m}{a^n}$?
4. What is the relationship between powers and roots in algebraic expressions?
5. How would this expression change if it were $\frac{(\sqrt{7})^4}{7}$?

Tip: When working with square roots and exponents, remember that $\sqrt{x} = x^{1/2}$, which helps simplify expressions efficiently.