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Which equation shows a valid step in solving [tex] \sqrt[3]{2x - 6} + \sqrt[3]{2x + 6} = 0 [/tex]?

A. [tex](\sqrt[3]{2x - 6})^2 = (\sqrt[3]{2x + 6})^2[/tex]

B. [tex](\sqrt[3]{2x - 6})^2 = (\sqrt[3]{2x + 6})^2[/tex]

C. [tex](\sqrt[3]{2x - 6})^3 = (\sqrt[3]{2x + 6})^3[/tex]

D. [tex](\sqrt[3]{2x - 6})^3 = (-\sqrt[3]{2x + 6})^3[/tex]

Sagot :

GinnyAnsweridn
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]
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