Example 31. Solve the equation

$$
\sqrt[3]{8+x}+\sqrt[3]{8-x}=1
$$
Solution. The domain of the equation is the set of all real numbers. Let $u=\sqrt[3]{8+x}, v=\sqrt[3]{8-x}$. Since $u^{3}+v^{3}=16$, for $u$ and $v$ we obtain the symmetric system of equations

$$
\left\{\begin{array}{c}
u+v=1 \\
u^{3}+v^{3}=16
\end{array}\right.
$$

solving which we find:

$$
\left\{\begin{array} { l } 
{ u _ { 1 } = \frac { 1 } { 2 } ( 1 + \sqrt { 2 1 } ) , } \\
{ v _ { 1 } = \frac { 1 } { 2 } ( 1 - \sqrt { 2 1 } ) , }
\end{array} \quad \left\{\begin{array}{l}
u_{2}=\frac{1}{2}(1-\sqrt{21}) \\
v_{2}=\frac{1}{2}(1+\sqrt{21})
\end{array}\right.\right.
$$

Thus, equation (20) is equivalent to the combination of two equations:

$$
\sqrt[3]{8+x}=\frac{1}{2}(1+\sqrt{21}), \quad \sqrt[3]{8+x}=\frac{1}{2}(1-\sqrt{21})
$$

Solving this combination, we find the set of all solutions to equation (20) - the numbers $3 \sqrt{21}$ and $(-3 \sqrt{21})$.

An equation of the form

$$
\sqrt[n]{a-f(x)}+\sqrt[n]{b+f(x)}=g(x)
$$

by introducing the variables

$$
u=\sqrt[n]{a-f(x)}, \quad v=\sqrt[n]{b+f(x)}
$$

is reduced to solving the system of algebraic equations

$$
\left\{\begin{aligned}
u+v & =g(x) \\
u^{n}+v^{n} & =a+b
\end{aligned}\right.
$$

72

## Ch. 2. EQUATIONS WITH ONE UNKNOWN