Question

(7+5root2)^1/3 + (7-5root2)^1/3 is

Answer 1

Answer:

$(7 + 5 \sqrt{2} ) {}^{ \frac{1}{3} } + (7 - 5 \sqrt{2} ) {}^{ \frac{1}{3} } \\ = \sqrt[3]{((7 + 5 \sqrt{2}) {}^{ \frac{1}{3} } + (7 - 5 \sqrt{2} ) {}^{ \frac{1}{3} }) {}^{3} } \\ = \sqrt[3]{7 + 5 \sqrt{2} + 7 - 5 \sqrt{2} + 3(7 + 5 \sqrt{2} )(7 - 5 \sqrt{2} )(7 + 5 \sqrt{2} + 7 - 5 \sqrt{2} )} \\ = \sqrt[3]{14 + 3( {7}^{2} - (5 \sqrt{2} ) {}^{2} ) \times 14 } \\ = \sqrt[3]{14 + 42(49 - 50)} \\ = \sqrt[3]{14 + 42 \times ( - 1)} \\ = \sqrt[3]{14 - 42} \\ = \sqrt[3]{ - 28} \\ = - 3.0365....$