X =
[tex](\sqrt{3{}^{5}}) - (\sqrt{2 {}^{6}})=
\sqrt{3*3*3*3*3}-\sqrt{2*2*2*2*2*2}=(\sqrt{3*3}*\sqrt{3*3}*\sqrt{3})-(\sqrt{2*2}-\sqrt{2*2}-\sqrt{2*2})=3*3*\sqrt{3}-(2*2*2)= 9\sqrt{3}+8[tex]
Y =
[tex](2 -\sqrt{3} ) {}^{2}=(2 -\sqrt{3} )(2 -\sqrt{3} ) = 2^{2}-2*2*\sqrt{3} + \sqrt{3^{2}}=7 -4\sqrt{3}[/tex]
On utilise l'identité remarquable (a-b)² = a² - 2ab² +b²
Z =
{ (2 - \sqrt{3} {}^{2}) } {}^{15} \: \times (7 + 4 \sqrt{3} ) {}^{15}
On applique : [tex]b^{a} \times c^{a} = (bc)^{a}
Donc [tex][(2 - \sqrt{3} {}^{2}) }(7 + 4 \sqrt{3} ) ]^{15}
On remarque que : [tex](2 - \sqrt{3} {}^{2}) = 2-3 = -1
[ (-1 ) }(7 + 4 \sqrt{3} ) ]^{15} = (-7 - 4 \sqrt{3} ) ^{15}
A =
( - 2 \sqrt{3} \: - 3 \sqrt{2)} (5 \sqrt{3} \: - 4 \sqrt{2} )
[- 2 \sqrt{3}\times5 \sqrt{3}] + [(- 2 \sqrt{3}) \times (- 4 \sqrt{2})] + [- 3 \sqrt{2} \times5 \sqrt{3}] + [- 3 \sqrt{2} \times (- 4 \sqrt{2})]
[- 2 \times 5 \times \sqrt{3}\times \sqrt{3}] + [-2\times-4\times \sqrt{3} \times \sqrt{2} ] + [- 3\times 5 \times \sqrt{2} \times \sqrt{3}] + [- 3 \times -4 \times \sqrt{2} \times \sqrt{2})]
(-30) + (8 \sqrt{6})+ (-15\sqrt{6}) + (24)
-6 -7\sqrt{6}
B =
( - 3 \sqrt{5} {}^{2} \: - 4 \sqrt{2} ) {}^{2} \: - 3 \sqrt{2} (4 \sqrt{5} - \sqrt{2} )
On utilise l'identité remarquable (a-b)² = a² - 2ab² +b²
[ (- 3 \sqrt{5} {}^{2})^{2} +(2\times- 3 \sqrt{5} {}^{2} \times - 4 \sqrt{2}) + (- 4 \sqrt{2} ) {}^{2} ] \: +[ - 3 \sqrt{2} \times 4 \sqrt{5} + 3 \sqrt{2}\times \sqrt{2} )]
[ (- 3 \times 5})^{2} +(2\times- 3 \times 5 \times - 4\times \sqrt{2})+ (- 4 \sqrt{2} ) {}^{2} ] \: +[ - 3 \times 4 \times\sqrt{5}\times\sqrt{2} + 3\times \sqrt{2}\times \sqrt{2} )]
[ 225 +(120\sqrt{2})+ 32 ] \: +[ -12\sqrt{10} + 6]
263 + 120\sqrt{2} -12\sqrt{10}
