Let's solve the problem step by step.

The expression given is $\sqrt{32x^6}$. We want to simplify it.

Step 1: Break down the square root

The square root can be separated as: $\sqrt{32x^6} = \sqrt{32} \cdot \sqrt{x^6}.$

Step 2: Simplify $\sqrt{32}$

The number $32$ can be factored as $32 = 16 \cdot 2$, so: $\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}.$

Step 3: Simplify $\sqrt{x^6}$

The square root of $x^6$ can be written as: $\sqrt{x^6} = x^3, \quad \text{because } \sqrt{x^{2n}} = x^n \text{ for even powers}.$

Step 4: Combine the results

Now combine the simplified terms: $\sqrt{32x^6} = 4\sqrt{2} \cdot x^3 = 4x^3\sqrt{2}.$

Final Answer:

The equivalent expression is A. $4x^3\sqrt{2}$.

Let me know if you'd like more clarification!

Related Questions:

1. How can you simplify $\sqrt{50x^4}$?
2. What is the general rule for simplifying square roots of variables with exponents?
3. Why is $\sqrt{16 \cdot 2}$ split into $\sqrt{16} \cdot \sqrt{2}$?
4. How do you handle cube roots in expressions like $\sqrt[3]{64x^9}$?
5. What is the relationship between exponents and roots?

Tip:

Always factorize the number under the square root into a perfect square and simplify the variables separately!