Question

Evaluate ${\left(3+\sqrt{2}\right)}^5-{\left(3-\sqrt{2}\right)}^5$

Answer 1

$(3+\sqrt{2}{)}^5-(3-\sqrt{2}{)}^5$

${}^5{C}_0{3}^5\left(\sqrt{2}{)}^0{+}^5{C}_1{3}^4\right(\sqrt{2}{)}^1{+}^5{C}_2{3}^3(\sqrt{2}{)}^2$

${+}^5{C}_3{3}^2\left(\sqrt{2}{)}^3{+}^5{C}_{4}3\right(\sqrt{2}{)}^4{+}^5{C}_5\left(3\right)(\sqrt{2}{)}^5$

$-{(}^5{C}_0{3}^5\left(\sqrt{2}\right){)}^0{-}^5{C}_1{3}^4\sqrt{2}{+}^5{C}_2{3}^3\left(\sqrt{2}{)}^2{-}^5{C}_3{3}^2\right({\sqrt{2}}^3{+}^5{C}_4{3}^1\left(\sqrt{2}\right){)}^2{-}^5{C}_5{3}^0(\sqrt{2}{)}^5$

$2{(}^5{C}_4{3}^4\sqrt{2}{+}^5{C}_2{3}^{2}2\sqrt{2}{+}^5{C}_{5}4\sqrt{2}$

$2\left(\right(5\left)\right(81)\sqrt{2}+10\times 9\times 2\sqrt{2}+4\sqrt{2})$

$810\sqrt{2}(180\sqrt{2}+4\sqrt{2})2$

$1178\sqrt{2}$