Let's solve the equation \(\sqrt[3]{5x - 2} - \sqrt[3]{4x} = 0\) step-by-step and examine which operations are performed in the process.
Given equation:
[tex]\[\sqrt[3]{5x - 2} - \sqrt[3]{4x} = 0\][/tex]
1. Isolate one of the cube roots:
[tex]\[\sqrt[3]{5x - 2} = \sqrt[3]{4x}\][/tex]
2. Cube both sides to eliminate the cube roots:
[tex]\[(\sqrt[3]{5x - 2})^3 = (\sqrt[3]{4x})^3\][/tex]
3. Simplify the result of cubing both sides:
[tex]\[5x - 2 = 4x\][/tex]
4. Solve the resulting linear equation:
[tex]\[5x - 4x = 2\][/tex]
[tex]\[x = 2\][/tex]
We have now solved the equation, obtaining \(x = 2\).
The crucial step where an operation was applied to manipulate the equation was when we cubed both sides. Cubing both sides allowed us to eliminate the cube roots and solve the equation.
Given the choices:
- Squaring both sides once
- Squaring both sides twice
- Cubing both sides once
- Cubing both sides twice
The correct choice is:
Cubing both sides once
