Let

25+22\sqrt{2} = (a+b\sqrt{2})^3

25-22\sqrt{2} = (a-b\sqrt{2})^3

in which a and b are positive integers

Addition of the two equations gives

(a+b\sqrt{2})^3+(a-b\sqrt{2})^3 = 50

Factoring the LHS using sum of cubes formula.

2a[(a+b\sqrt{2})^2+(a-b\sqrt{2})^2-a^2+2b^2)=50

a(2a^2+2b^2\cdot 2 - a^2+2b^2) = 25

Multiplying the two equations gives

(25+22\sqrt{2})(25-22\sqrt{2}) = (a+b\sqrt{2})^3 (a-b\sqrt{2})^3

Simplify the equation using difference of square formula

(a^2 - b^2\cdot 2)^3 = 25^2 - 22^2\cdot 2=-343

Taking cube root gives

A system of equations is obtained from (1) and (2)

\begin{cases} a^3 + 6ab^2 &=25 \\ a^2-2b^2 &=-7 \end{cases}

Solving the equation yields

a = 1, b=2

Therefore,

The cube root of

\sqrt[3]{25+22\sqrt{2}}=\sqrt[3]{(1+2\sqrt{2})^3}=1+2\sqrt{2}

\sqrt[3]{25-22\sqrt{2}}=\sqrt[3]{(1-2\sqrt{2})^3}=1-2\sqrt{2}