To solve the equation [tex]\(\sqrt[3]{2x - 6} + \sqrt[3]{2x + 6} = 0\)[/tex], we need to isolate each term and then manipulate the equation appropriately. Here is the step-by-step process:
1. Isolate one of the cube roots:
[tex]\[
\sqrt[3]{2x - 6} = -\sqrt[3]{2x + 6}
\][/tex]
2. Eliminate the cube roots by cubing both sides:
[tex]\[
(\sqrt[3]{2x - 6})^3 = (-\sqrt[3]{2x + 6})^3
\][/tex]
This gives us:
[tex]\[
2x - 6 = -(\sqrt[3]{2x + 6})^3
\][/tex]
However, we know that cubing a negative term will introduce a negative sign on the right side:
[tex]\[
2x - 6 = - (2x + 6)
\][/tex]
So the valid step we are looking for is:
[tex]\[
(\sqrt[3]{2x - 6})^3 = (-\sqrt[3]{2x + 6})^3
\][/tex]
Therefore, the correct equation from the given options is:
[tex]\[
(\sqrt[3]{2x - 6})^3 = (-\sqrt[3]{2x + 6})^3
\][/tex]
