Let

90+34\sqrt{7} = (a+b\sqrt{7})^3

90-34\sqrt{7} = (a-b\sqrt{7})^3

in which a and b are positive integers

Addition of the two equations gives

(a+b\sqrt{7})^3+(a-b\sqrt{7})^3 = 180

Factoring the LHS using sum of cubes formula.

2a[(a+b\sqrt{7})^2+(a-b\sqrt{7})^2-a^2+7b^2)=180

a(2a^2+2b^2\cdot 7 - a^2+7b^2) = 90

Multiplying the two equations gives

(90+34\sqrt{7})(90-34\sqrt{7}) = (a+b\sqrt{7})^3 (a-b\sqrt{7})^3

Simplify the equation using difference of square formula

(a^2 - b^2\cdot 7)^3 = 90^2 - 34^2\cdot 7=8

Taking cube root gives

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

\begin{cases} a^3 + 21ab^2 &=90 \\ a^2-7b^2 &=2 \end{cases}

Solving the equation yields

a = 3, b=1

Therefore,

The cube root of

\sqrt[3]{90+34\sqrt{7}}=\sqrt[3]{(3+\sqrt{7})^3}=3+\sqrt{7}

\sqrt[3]{90-34\sqrt{7}}=\sqrt[3]{(3-\sqrt{7})^3}=3-\sqrt{7}