Ecrire les expressions sous la formes a racine de b où a et b sont des entiers, avec b le plus petit possible.

A= 5 racine 18 +3 racine 2 - 4 racine 8 + 7 racine 50=
B=5 racine 72 - 6 racine 32 - 9 racine 2 +racine 98=
C= racine 20 - racine 15au carré *5 + 2 racine 45=
D= 3 racine 72 - 5 racine 2 + 7 racine 128=


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2014-06-14T09:16:41+02:00
Bonjour,

A= 5 racine 18 +3 racine 2 - 4 racine 8 + 7 racine 50=

A= 5\sqrt{18} +3\sqrt{2} - 4\sqrt{8} + 7\sqrt{50}\\\\A= 5\sqrt{9\times2} +3\sqrt{2} - 4\sqrt{4\times2} + 7\sqrt{25\times2}\\\\A= 5\sqrt{9}\times\sqrt{2} +3\sqrt{2} - 4\sqrt{4}\times\sqrt{2} + 7\sqrt{25}\times\sqrt{2}\\\\A= 5\times3\times\sqrt{2} +3\sqrt{2} - 4\times2\times\sqrt{2} + 7\times5\times\sqrt{2}\\\\A= 15\sqrt{2} +3\sqrt{2} -8\sqrt{2} + 35\sqrt{2}\\\\\boxed{A= 45\sqrt{2}}

B=5 racine 72 - 6 racine 32 - 9 racine 2 +racine 98=

B=5\sqrt{72} - 6\sqrt{32} - 9\sqrt{2} +\sqrt{98}\\\\B=5\sqrt{36\times2} - 6\sqrt{16\times2} - 9\sqrt{2} +\sqrt{49\times2}\\\\B=5\sqrt{36}\sqrt{2} - 6\sqrt{16}\sqrt{2} - 9\sqrt{2} +\sqrt{49}\sqrt{2}\\\\B=5\times6\sqrt{2} - 6\times4\sqrt{2} - 9\sqrt{2} +7\sqrt{2}\\\\B=30\sqrt{2} - 24\sqrt{2} - 9\sqrt{2} +7\sqrt{2}\\\\\boxed{B=4\sqrt{2}}

C= racine 20 - racine 15au carré *5 + 2 racine 45=

C=\sqrt{20} - \sqrt{15^2\times5} + 2\sqrt{45}\\\\C=\sqrt{4\times5} - 15\times\sqrt{5} + 2\sqrt{9\times5}\\\\C=\sqrt{4}\times\sqrt{5} - 15\times\sqrt{5} + 2\sqrt{9}\times\sqrt{5}\\\\C=2\times\sqrt{5} - 15\times\sqrt{5} + 2\times3\times\sqrt{5}\\\\C=2\sqrt{5} - 15\sqrt{5} + 6\sqrt{5}\\\\\boxed{C=-7\sqrt{5}}

D= 3 racine 72 - 5 racine 2 + 7 racine 128=

D= 3\sqrt{72} - 5\sqrt{2} + 7\sqrt{128}\\\\D= 3\sqrt{36\times2} - 5\sqrt{2} + 7\sqrt{64\times2}\\\\D= 3\sqrt{36}\times\sqrt{2} - 5\sqrt{2} + 7\sqrt{64}\times\sqrt{2}\\\\D= 3\times6\times\sqrt{2} - 5\sqrt{2} + 7\times8\times\sqrt{2}\\\\D=18\sqrt{2} - 5\sqrt{2} + 56\sqrt{2}\\\\\boxed{D=69\sqrt{2}}