Solve the equation $x+a^3=\sqrt[3]{a-x}$ where $a$ is real.
If one wants to do things in haste and quickly solve the above equation for $x$ without thinking much, one would definitely raise both sides of the equation to the third power to get rid of the cube root:
$x+a^3=\sqrt[3]{a-x}$
$(x+a^3)^3=(\sqrt[3]{a-x})^3$