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\(a,A=\left(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}\right)^2\)
\(=6-2\sqrt{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}\)
\(=2\)
\(b,B=\sqrt{227-30\sqrt{2}}+\sqrt{123+22\sqrt{2}}\)
\(=\sqrt{\left(15-\sqrt{2}\right)^2+\left(11+\sqrt{2}\right)^2}\)
\(=26\)
hơi tắt
a)
=\(\sqrt{15^2-2\cdot15\cdot\sqrt{2}+2}+\sqrt{11^2+2\cdot11\cdot\sqrt{2}+2}\)
=\(\sqrt{\left(15-\sqrt{2}\right)^2}+\sqrt{\left(11+\sqrt{2}\right)}^2\)
=\(15-\sqrt{2}+11+\sqrt{2}\)
=26
c)
=\(\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{2}}\left(\sqrt{5}+2\right)\)
=\(\dfrac{\left(\sqrt{3}-1\right)\left(\sqrt{5}+\sqrt{2}\right)}{\sqrt{2}}\)
\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
= \(\sqrt{13+30\sqrt{2+\sqrt{\left(1+2\sqrt{2}\right)^2}}}\)= \(\sqrt{13+30\sqrt{\left(1+\sqrt{2}\right)^2}}\)
= \(\sqrt{43\:+30\sqrt{2}}\) = \(\sqrt{(25+2×5×3\sqrt{2}+18}\) = \(5\:+3\sqrt{2}\)
\(\sqrt{227-30\sqrt{2}}+\sqrt{123+22\sqrt{2}}=\sqrt{225-30\sqrt{2}+2}+\sqrt{121+22\sqrt{2}+2}=\sqrt{15^2-15.2.\sqrt{2}+\left(\sqrt{2}\right)^2}+\sqrt{11^2+11.2\sqrt{2}+\left(\sqrt{2}\right)^2}=\sqrt{\left(15-\sqrt{2}\right)^2}+\sqrt{\left(11+\sqrt{2}\right)^2}=15-\sqrt{2}+11+\sqrt{2}\left(do:15-\sqrt{2}>0;11+\sqrt{2}>0\right)=26\)
\(\sqrt{227-30\sqrt{2}}+\sqrt{123+22\sqrt{2}}\)
=\(\sqrt{225+2.15.\sqrt{2}+2}+\sqrt{121+2.11\sqrt{2}+2}\)
=\(\sqrt{\left(15+\sqrt{2}\right)^2}+\sqrt{\left(11+\sqrt{2}\right)^2}\)
=\(15+\sqrt{2}+11+\sqrt{2}\)
=\(26+2\sqrt{2}\)
\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)=\(\sqrt{13+30\sqrt{2+\sqrt{\left(\sqrt{8}+1\right)^2}}}\)
=\(\sqrt{13+30\sqrt{2+\sqrt{8}+1}}\)=\(\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)=\(\sqrt{13+30\sqrt{2}+30}\)
=\(\sqrt{43+30\sqrt{2}}\)=\(\sqrt{\left(5+3\sqrt{2}\right)^2}\)=\(5+3\sqrt{2}\)
\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}.\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}\right)^2+2.2\sqrt{2}+1}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)
\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
\(=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{13+30\sqrt{2}+30}=\sqrt{43+30\sqrt{2}}\)
a) \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\\ =\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\\ =\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\\ =\sqrt{13+30\left(\sqrt{2}+1\right)}\\ =\sqrt{13+30\sqrt{2}+30}\\ =\sqrt{43+30\sqrt{2}}\\ =\sqrt{\left(3\sqrt{2}+5\right)^2}\\ =3\sqrt{2}+5\)
b) \(\sqrt{227-30\sqrt{2}}+\sqrt{123+22\sqrt{2}}\\ =\sqrt{225-2\cdot15\sqrt{2}+2}+\sqrt{121+2\cdot11\sqrt{2}+2}\\ =\sqrt{\left(15-\sqrt{2}\right)^2}+\sqrt{\left(11+\sqrt{2}\right)^2}\\ =15-\sqrt{2}+11+\sqrt{2}\\ =26\)