Question

3. Dacă a, b apartine N,
$\sqrt{46 - 16 \sqrt{6} } = a \sqrt{2} - b \sqrt{3}$
atunci a - b este:​

Answer 1

 

$\displaystyle\bf\\Enuntul~problemei~este:~~\sqrt{46-16\sqrt{6}}=a\sqrt{2}-b\sqrt{3}\\\\Enuntul~problemei~contine~o~greseala.\\\\\textbf{Voi rezolva problema considerand ca in loc de 46 este 44}\\\\16\sqrt{6}=2\times4\times2\times\sqrt{2}\sqrt{3}=2\times4\sqrt{2}\times2\sqrt{3}\\\\Verificare:\\\\\Big(4\sqrt{2}\Big)^2+\Big(2\sqrt{3}\Big)^2=32+12=44~~~Corect ~!$

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$\displaystyle\bf\\Rezolvare\\44=32+12\\\sqrt{44-16\sqrt{6}}=\\\\=\sqrt{32+12-2\times4\sqrt{2}\times2\sqrt{3}}=\\\\=\sqrt{32-2\times4\sqrt{2}\times2\sqrt{3}+12}}=\\\\=\sqrt{\Big(4\sqrt{2}\Big)^2-2\times4\sqrt{2}\times2\sqrt{3}+\Big(2\sqrt{3}\Big)^2}}=\\\\=\sqrt{\Big(4\sqrt{2}-2\sqrt{3}\Big)^2}}=\\\\=\Big|4\sqrt{2}-2\sqrt{3}\Big|=\\\\=\boxed{\bf4\sqrt{2}-2\sqrt{3}}\\..............\\4\sqrt{2}-2\sqrt{3}=a\sqrt{2}-b\sqrt{3}\\\\\implies~~\boxed{\bf~a=4~~si~~b=2}\\\\\implies~~~\boxed{\bf~a-b=4-2=2}$