+26393 Find all the solutions to
\(\sqrt[3]{ \sqrt[3]{5\sqrt{2}+x}+ \sqrt[3]{5\sqrt{2}-x} }=\sqrt{2}\)
\(\small{ \begin{array}{|rcll|} \hline \sqrt[3]{ \sqrt[3]{5\sqrt{2}+x}+ \sqrt[3]{5\sqrt{2}-x} } &=& \sqrt{2} \quad & | \quad \text{cube both sides}\\ \sqrt[3]{5\sqrt{2}+x}+ \sqrt[3]{5\sqrt{2}-x} &=& \left(\sqrt{2}\right)^3 \quad & | \quad \text{cube both sides} \\ \left( \sqrt[3]{5\sqrt{2}+x}+ \sqrt[3]{5\sqrt{2}-x}\right)^3 &=& \left(\sqrt{2}\right)^9 \\ \boxed{(a+b)^3 = a^3+b^3+3ab(a+b) } \\ 5\sqrt{2}+x + 5\sqrt{2}-x +3\sqrt[3]{(5\sqrt{2}+x)(5\sqrt{2}-x)}( \underbrace{ \sqrt[3]{5\sqrt{2}+x}+ \sqrt[3]{5\sqrt{2}-x}}_{=\left(\sqrt{2}\right)^3} ) &=& \left(\sqrt{2}\right)^9 \\ 10\sqrt{2} +3\sqrt[3]{25*2-x^2}\left(\sqrt{2}\right)^3 &=& \left(\sqrt{2}\right)^9 \\ 3\sqrt[3]{50-x^2} &=& \dfrac{\left(\sqrt{2}\right)^9 -10\sqrt{2} } {\left(\sqrt{2}\right)^3} \\ 3\sqrt[3]{50-x^2} &=& \dfrac{\sqrt{2}\left( \left(\sqrt{2}\right)^8-10 \right) } {\sqrt{2} \left(\sqrt{2}\right)^2 } \\ 3\sqrt[3]{50-x^2} &=& \dfrac{\left(\sqrt{2}\right)^8-10} {\left(\sqrt{2}\right)^2 } \\ 3\sqrt[3]{50-x^2} &=& \dfrac{16-10} {4} \\ 3\sqrt[3]{50-x^2} &=& 3 \\ \sqrt[3]{50-x^2} &=& 1 \quad & | \quad \text{cube both sides} \\ 50-x^2 &=& 1 \\ x^2 &=& 49 \\ x &=& \pm 7 \\ \hline \end{array} }\)
Solutions:
\(x=-7 \\ x=7\)
