To solve the radical equation \( \sqrt[5]{400 - 6x^3} = 4 \), we first eliminate the fifth root by raising both sides of the equation to the fifth power:
\[ 400 - 6x^3 = 4^5 \]
Calculating \( 4^5 \):
\[ 4^5 = 1024 \]
So now our equation becomes:
\[ 400 - 6x^3 = 1024 \]
Next, we can isolate the \( -6x^3 \) term by subtracting 400 from both sides:
\[ -6x^3 = 1024 - 400 \]
Calculating the right side:
\[ -6x^3 = 624 \]
Now, divide both sides by -6:
\[ x^3 = -\frac{624}{6} = -104 \]
Next, we need to take the cube root of both sides:
\[ x = \sqrt[3]{-104} \]
This gives us the solution:
\[ x = -\sqrt[3]{104} \]
Now we can check the answer by verifying the options provided. The closest match to this solution looks to be \( x = -213^{\sqrt{3}} \), or something similar to that. However, I would specifically clarify that \( -\sqrt[3]{104} \) is the most accurate answer to the given equation. The response options seem a bit different but based on cube roots, the answer is:
\[ x = -\sqrt[3]{104} \]
This was computed as a fundamental rule for solving radical equations. Please check for correct formatting and possible typos in your provided responses.