Question
Find cube root of \sqrt[3]{90+34\sqrt{7}}
Find cube root of \sqrt[3]{90+34\sqrt{7}}
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}