Question Number 12361 by Joel576 last updated on 20/Apr/17
$$\left(\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{3}}\:+\mathrm{2}\:+\:\sqrt{\mathrm{5}}\right)\left(−\sqrt{\mathrm{2}}\:+\:\sqrt{\mathrm{3}}\:+\:\mathrm{2}\:−\:\sqrt{\mathrm{5}}\right)\left(\sqrt{\mathrm{10}}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\right) \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{without}\:\mathrm{solving}\:\mathrm{them}\:\mathrm{manually}? \\ $$
Answered by ajfour last updated on 20/Apr/17
 =[b^2 +c^2 −a^2 −d^2 +2(bc−ad)](ad+bc) =[b^2 +c^2 −a^2 −d^2 ](ad+bc)+2(b^2 c^2 −a^2 d^2 ) =0+2(12−10) =4 .](Q12362.png)
$$\left({a}+{b}+{c}+{d}\right)\left({b}+{c}−{a}−{d}\right)\left({ad}+{bc}\right) \\ $$$$=\left[\left({b}+{c}\right)^{\mathrm{2}} −\left({a}+{d}\right)^{\mathrm{2}} \right]\left({ad}+{bc}\right) \\ $$$$=\left[{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} −{d}^{\mathrm{2}} +\mathrm{2}\left({bc}−{ad}\right)\right]\left({ad}+{bc}\right) \\ $$$$=\left[{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} −{d}^{\mathrm{2}} \right]\left({ad}+{bc}\right)+\mathrm{2}\left({b}^{\mathrm{2}} {c}^{\mathrm{2}} −{a}^{\mathrm{2}} {d}^{\mathrm{2}} \right) \\ $$$$=\mathrm{0}+\mathrm{2}\left(\mathrm{12}−\mathrm{10}\right)\:=\mathrm{4}\:. \\ $$