To simplify the given radical expression, we have:

$\sqrt{32x^5}$

Step-by-Step Simplification:

1. Factor the number 32 under the square root: $32 = 2^5$ So, the expression becomes: $\sqrt{32x^5} = \sqrt{2^5 x^5}$

2. Use the property of square roots: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ This allows us to split the square root into two parts: $\sqrt{2^5} \cdot \sqrt{x^5}$

3. Simplify each square root:

   • For $\sqrt{2^5}$, we can write it as: $\sqrt{2^5} = \sqrt{2^4 \cdot 2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}$
   • For $\sqrt{x^5}$, we split it similarly: $\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2 \cdot \sqrt{x}$

4. Combine the simplified results: Now, putting everything together: $4\sqrt{2} \cdot x^2 \cdot \sqrt{x}$

5. Final simplified expression: $4x^2\sqrt{2x}$

Thus, the simplified form of $\sqrt{32x^5}$ is:

$4x^2\sqrt{2x}$

Would you like further details or have any other questions?

Here are 5 related questions:

1. How would you simplify $\sqrt{50x^7}$?
2. What happens when simplifying cube roots of variables, e.g., $\sqrt[3]{64x^6}$?
3. How does the simplification of $\sqrt{75x^3}$ compare to this?
4. Can all radical expressions with powers of variables be simplified in the same way?
5. What role do the properties of exponents play in simplifying radical expressions?

Tip: Always try to break down both the numerical and variable parts separately when simplifying a square root.