\(P=\sqrt[3]{22\sqrt{2}+25}-\sqrt[3]{22\sqrt{2}-25}\)
\(=\sqrt[3]{22\sqrt{2}+25}+\sqrt[3]{\left(-1\right)\cdot\left(22\sqrt{2}-25\right)}\)
\(=\sqrt[3]{25+22\sqrt{2}}+\sqrt[3]{25-22\sqrt{2}}\)
=>\(P^3=25+22\sqrt{2}+25-22\sqrt{2}+3\cdot\sqrt[3]{\left(25-22\sqrt{2}\right)\left(25+22\sqrt{2}\right)}\cdot P\)
=>\(P^3=50+3\cdot\sqrt[3]{625-968}\cdot P\)
=>\(P^3=50+3\cdot\left(-7\right)\cdot P\)
=>\(P^3+21P-50=0\)
=>\(P^3-2P^2+2P^2-4P+25P-50=0\)
=>\(\left(P-2\right)\left(P^2+2P+25\right)=0\)
=>P-2=0
=>P=2
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