Question

Вычеслите √(14 + 6 × √5) – √(9 – 4 × √5)

Answer 1

$\sqrt{14+6\sqrt{5}}=\sqrt{9+6\sqrt{5}+5}=\\\\\sqrt{3^2+2*3*\sqrt{5}+(\sqrt{5})^2}=\\\\\sqrt{(3+\sqrt{5})^2=\\\\|3+\sqrt{5}|=3+\sqrt{5}$
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$\sqrt{9-4\sqrt{5}}=\sqrt{4-4\sqrt{5}+5}=\\\\\sqrt{2^2-2*2*\sqrt{5}+(\sqrt{5})^2}=\\\\\sqrt{(2-\sqrt{5})^2}=\\\\|2-\sqrt{5}|=\sqrt{5}-2$
так как
$2-\sqrt{5}=\sqrt{4}-\sqrt{5}<0$
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$\sqrt{14+6\sqrt{5}}-\sqrt{9-4\sqrt{5}}=(3+\sqrt{5})-(\sqrt{5}-2)=\\\\3+\sqrt{5}-\sqrt{5}+2=5$
ответ: 5