Question

4 / (√2 + √(6 + 4 × √(2))
избавиться от иррациональности в знаменатели

Answer 1

$frac{4}{ sqrt{2} + sqrt{6 + 4 sqrt{2} } } = frac{4}{ sqrt{2} + sqrt{4 + 4 sqrt{2} + 2} } = \ = frac{4}{ sqrt{2} + sqrt{(2 + sqrt{2}) ^{2} } } = frac{4}{ sqrt{2} + 2 + sqrt{2} } = frac{4}{2 sqrt{2} + 2 } = \ = frac{4}{2( sqrt{2} + 1)} = frac{2}{ sqrt{2} + 1} = frac{2( sqrt{2} - 1)}{( sqrt{2} + 1)( sqrt{2} - 1)} = frac{2( sqrt{2} - 1)}{2 - 1} = 2( sqrt{2} - 1) = \ = 2 sqrt{2 } - 2$

Answer 2

$frac{4}{sqrt{2}+sqrt{6+4sqrt{2}}}=frac{4}{sqrt{2}+sqrt{4+4sqrt{2}+2}}=frac{4}{sqrt{2}+sqrt{2^2+2*2sqrt{2}+(sqrt{2})^2}}= \ \ =frac{4}{sqrt{2}+sqrt{(2+sqrt{2})^2}}=frac{4}{sqrt{2}+|2+sqrt{2}|}}=frac{4}{sqrt{2}+2+sqrt{2}}}=frac{4}{2(sqrt{2}+1)}=\ \ =frac{2(sqrt{2}-1)}{(sqrt{2}+1)(sqrt{2}-1)}=frac{2(sqrt{2}-1)}{2-1}=2(sqrt{2}-1)$