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\(\sqrt{17-12\sqrt{2}}+\sqrt{17+12\sqrt{2}}=\sqrt{17-2\sqrt{72}}+\sqrt{17+2\sqrt{72}}\)
=\(\sqrt{9-2\sqrt{9}.\sqrt{8}+8}+\sqrt{9+2\sqrt{9}.\sqrt{8}+8}\)
=\(\sqrt{\left(\sqrt{9}-\sqrt{8}\right)^2}+\sqrt{\left(\sqrt{9}+\sqrt{8}\right)^2}\)
=\(\sqrt{9}-\sqrt{8}+\sqrt{9}+\sqrt{8}=3+3=6\)
\(\sqrt{17-12\sqrt{2}}+\sqrt{17+12\sqrt{2}}=\sqrt{17-12\sqrt{72}}+\sqrt{17+12\sqrt{72}}=\sqrt{9-2\sqrt{9}.\sqrt{8}+8}+\sqrt{9+2\sqrt{9}.\sqrt{8}+8}=6\)
\(a,\sqrt{\sqrt{17+12\sqrt{2}}}\)
\(=\sqrt{\sqrt{8+12\sqrt{2}+9}}\)
\(=\sqrt{\sqrt{\left[2\sqrt{2}+3\right]^2}}\)
\(=\sqrt{2\sqrt{2}+3}\)
\(=\sqrt{1+2\sqrt{2}+2}\)
\(=\sqrt{\left[1+\sqrt{2}\right]^2}\)
\(=1+\sqrt{2}\)
\(b,\sqrt{4+2\sqrt{3}}-\sqrt{21-12\sqrt{3}}\)
\(=\sqrt{3+2\sqrt{3}+1}-\sqrt{12-12\sqrt{3}+9}\)
\(=\sqrt{\left[1+\sqrt{3}\right]^2}-\sqrt{\left[2\sqrt{3}-3\right]^2}\)
\(=\left(1+\sqrt{3}\right)-\left(2\sqrt{3}-3\right)\)
\(=1+\sqrt{3}-2\sqrt{3}+3\)
\(=4-\sqrt{3}\)
chúc bn học tốt
\(\sqrt{242}.\sqrt{26}.\sqrt{130}.\sqrt{0,9}-\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)\)
\(=\sqrt{121}.\sqrt{2}.\sqrt{2}.\sqrt{13}.\sqrt{13}.\sqrt{10}.\sqrt{0,9}-\left(2-1\right)\)
\(=11.2.13.\sqrt{9}-1=286.3-1=857\)
\(\frac{3-\sqrt{6}}{\sqrt{12}-\sqrt{8}}-\frac{\sqrt{15}-\sqrt{5}}{2\sqrt{12}-4}+\frac{\sqrt{17-4\sqrt{15}}}{4}\)
\(=\frac{\sqrt{3}\left(\sqrt{3}-\sqrt{2}\right)}{2\left(\sqrt{3}-\sqrt{2}\right)}-\frac{\sqrt{5}\left(\sqrt{3}-1\right)}{4\left(\sqrt{3}-1\right)}+\frac{\sqrt{\left(2\sqrt{3}-\sqrt{5}\right)^2}}{4}\)
\(=\frac{\sqrt{3}}{2}-\frac{\sqrt{5}}{4}+\frac{2\sqrt{3}-\sqrt{5}}{4}\)
\(=\sqrt{3}-\frac{\sqrt{5}}{4}\)
\(\sqrt{6-4\sqrt{2}}\)\(+\sqrt{22-12\sqrt{2}}\)
\(=\sqrt{4-4\sqrt{2}+2}\)\(+\sqrt{18-12\sqrt{2}+4}\)
\(=\sqrt{\left(2-\sqrt{2}\right)^2}\)\(+\sqrt{\left(2-3\sqrt{2}\right)^2}\)
\(=2-\sqrt{2}+3\sqrt{2}-2\)
\(=\left(2-2\right)+\left(-\sqrt{2}+3\sqrt{2}\right)\)
\(=0+2\sqrt{2}\)\(=2\sqrt{2}\)
\(\sqrt{17-12\sqrt{2}}\)\(+\sqrt{9+4\sqrt{2}}\)
\(=\sqrt{\left(3-2\sqrt{2}\right)^2}\)\(+\sqrt{\left(2\sqrt{2}+1\right)^2}\)
\(=\left|3-2\sqrt{2}\right|\)\(+\left|2\sqrt{2}+1\right|\)
\(=3-2\sqrt{2}\)\(+2\sqrt{2}+1\)
\(=\left(3+1\right)+\left(-2\sqrt{2}+2\sqrt{2}\right)\)
\(=4+0=4\)
\(A=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(=|2+\sqrt{3}|-|2-\sqrt{3}|\)
\(=2+\sqrt{3}-2+\sqrt{3}\)
\(=2\sqrt{3}\)
\(B=\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}\)
\(=\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}\)
\(=|3+\sqrt{2}|-|3-\sqrt{2}|\)
\(=3+\sqrt{2}-3+\sqrt{2}\)
\(=2\sqrt{2}\)
\(C=\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}+\sqrt{\left(3-2\sqrt{2}\right)^2}\)
\(=|3+2\sqrt{2}|+|3-2\sqrt{2}|\)
\(=3+2\sqrt{2}+3-2\sqrt{2}\)
\(=6\)
\(D=\sqrt{9+4\sqrt{5}}-\sqrt{9-4\sqrt{5}}\)
\(=\sqrt{\left(2+\sqrt{5}\right)^2}-\sqrt{\left(2-\sqrt{5}\right)^2}\)
\(=|2+\sqrt{5}|-|2-\sqrt{5}|\)
\(=2+\sqrt{5}-\sqrt{5}+2\)
\(=4\)
\(E=\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}\)
\(=\sqrt{\left(1+\sqrt{5}\right)^2}-\sqrt{\left(1-\sqrt{5}\right)^2}\)
\(=|1+\sqrt{5}|-|1-\sqrt{5}|\)
\(=1+\sqrt{5}-\sqrt{5}+1\)
\(=2\)
\(A=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\)
\(A=\sqrt{3}+2+2-\sqrt{3}\)
A = 2 + 2
A = 4
\(B=\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}\)
\(B=\sqrt{2}+3+3-\sqrt{2}\)
B = 3 + 3
B = 6
\(C=\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}}\)
\(C=3+2\sqrt{2}+3-2\sqrt{2}\)
C = 3 + 3
C = 6
\(D=\sqrt{9+4\sqrt{5}}-\sqrt{9-4\sqrt{5}}\)
\(D=\sqrt{5}+2-\sqrt{5}+2\)
D = 2 + 2
D = 4
\(E=\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}\)
\(E=\sqrt{5}+1-\sqrt{5}+1\)
E = 1 + 1
E = 2
f, \(\sqrt{\sqrt{5}+\sqrt{3-\sqrt{29-12\sqrt{5}}}}=\sqrt{\sqrt{5}+\sqrt{3-\sqrt{\left(2\sqrt{5}-3\right)^2}}}=\sqrt{\sqrt{5}+\sqrt{3-2\sqrt{5}+3}}=\sqrt{\sqrt{5}+\sqrt{6-2\sqrt{5}}}=\sqrt{\sqrt{5}+\sqrt{\left(\sqrt{5}-1\right)^2}}=\sqrt{\sqrt{5}+\sqrt{5}-1}=\sqrt{2\sqrt{5}-1}\)
mik sửa lại câu f , tí nhé :
f , \(\sqrt{\sqrt{5}+\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)
Trả lời
\(\sqrt{17+12\sqrt{2}}=\sqrt{9+12\sqrt{2}+8}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}\)
\(=3+2\sqrt{2}\)
\(\sqrt{17+12\sqrt{2}}-\sqrt{17-12\sqrt{2}}\)
\(=\sqrt{3^2+2\cdot3\cdot2\sqrt{2}+\left(2\sqrt{2}\right)^2}-\sqrt{3^2-2\cdot3\cdot2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}-\sqrt{\left(3-2\sqrt{2}\right)^2}\)
\(=\left|3+2\sqrt{2}\right|-\left|3-2\sqrt{2}\right|\)
\(=3+2\sqrt{2}-3+2\sqrt{2}\)
\(=4\sqrt{2}\)
\(\sqrt{17+12\sqrt{2}}-\sqrt{17-12\sqrt{2}}\)
\(=\sqrt{3^2+2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}-\sqrt{3^2-2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}-\sqrt{\left(3-2\sqrt{2}\right)^2}\)
\(=\left|3+2\sqrt{2}\right|-\left|3-2\sqrt{2}\right|=\left(3+2\sqrt{2}\right)-\left(3-2\sqrt{2}\right)\)
\(=3+2\sqrt{2}-3+2\sqrt{2}=4\sqrt[]{2}\)