#### Answer

$x=3$

#### Work Step by Step

Cube both sides of the equation to obtain:
$$((14-2x)^{2/3})^3=4^3
\\(14-2x)^2=64$$
Take the square root of both sides of the equation to obtain:
$$\sqrt{(14-2x)^2}=\pm\sqrt{64}
\\14-2x=\pm8$$
Subtract $14$ to both sides of the equation to obtain:
$$14-2x-14 =\pm8 - 14
\\-2x = -14\pm8$$
Divide $-2$ to both sides of the equation to obtain:
$$\dfrac{-2x}{-2}=\dfrac{-14\pm8}{-2}
\\x=7 \mp 4
\\x_1=7-4=3
\\x_2=7+4=11$$
However, when $x=11$, the given equation becomes
$$(14-2\cdot11)^{2/3} = 4
\\(14-22)^{2/3}=4
\\(-11)^{2/3}\ne4$$
This means that $11$ is an extraneous solution.
Thus, the solution to the given equation is $3$.