\(\sqrt{17+12\sqrt{2}}-\sqrt{17-12\sqrt{2}}\)

\(=\sqrt{3^2+2\cdot3\cdot2\sqrt{2}+\left(2\sqrt{2}\right)^2}-\sqrt{3^2-2\cdot3\cdot2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)

\(=\sqrt{\left(3+2\sqrt{2}\right)^2}-\sqrt{\left(3-2\sqrt{2}\right)^2}\)

\(=\left|3+2\sqrt{2}\right|-\left|3-2\sqrt{2}\right|\)

\(=3+2\sqrt{2}-3+2\sqrt{2}\)

\(=4\sqrt{2}\)

\(\sqrt{17+12\sqrt{2}}-\sqrt{17-12\sqrt{2}}\)

\(=\sqrt{3^2+2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}-\sqrt{3^2-2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)

\(=\sqrt{\left(3+2\sqrt{2}\right)^2}-\sqrt{\left(3-2\sqrt{2}\right)^2}\)

\(=\left|3+2\sqrt{2}\right|-\left|3-2\sqrt{2}\right|=\left(3+2\sqrt{2}\right)-\left(3-2\sqrt{2}\right)\)

\(=3+2\sqrt{2}-3+2\sqrt{2}=4\sqrt[]{2}\)

\(\dfrac{\sqrt{3-2\sqrt{2}}}{\sqrt{17-12\sqrt{2}}}-\dfrac{\sqrt{3+2\sqrt{2}}}{\sqrt{17+12\sqrt{2}}}\)

\(=\dfrac{\sqrt{\left(\sqrt{2}\right)^2-2.\sqrt{2}.1+1^2}}{\sqrt{3^2-2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}}-\dfrac{\sqrt{\left(\sqrt{2}\right)^2+2.\sqrt{2}.1+1^2}}{\sqrt{3^2+2.3.2\sqrt{2}+\left(2\sqrt{2}\right)^2}}\)

\(=\dfrac{\sqrt{\left(\sqrt{2}-1\right)^2}}{\sqrt{\left(3-2\sqrt{2}\right)^2}}-\dfrac{\sqrt{\left(\sqrt{2}+1\right)^2}}{\sqrt{\left(3+2\sqrt{2}\right)^2}}=\dfrac{\sqrt{2}-1}{3-2\sqrt{2}}-\dfrac{\sqrt{2}+1}{3+2\sqrt{2}}\)

\(=\dfrac{\sqrt{2}-1}{\left(\sqrt{2}-1\right)^2}+\dfrac{\sqrt{2}+1}{\left(\sqrt{2}+1\right)^2}=\dfrac{1}{\sqrt{2}-1}+\dfrac{1}{\sqrt{2}+1}\)

\(=\dfrac{\sqrt{2}+1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}-\dfrac{\sqrt{2}-1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}=\sqrt{2}+1-\sqrt{2}+1=2\)

\(a,\sqrt{\sqrt{17+12\sqrt{2}}}\)

\(=\sqrt{\sqrt{8+12\sqrt{2}+9}}\)

\(=\sqrt{\sqrt{\left[2\sqrt{2}+3\right]^2}}\)

\(=\sqrt{2\sqrt{2}+3}\)

\(=\sqrt{1+2\sqrt{2}+2}\)

\(=\sqrt{\left[1+\sqrt{2}\right]^2}\)

\(=1+\sqrt{2}\)

\(b,\sqrt{4+2\sqrt{3}}-\sqrt{21-12\sqrt{3}}\)

\(=\sqrt{3+2\sqrt{3}+1}-\sqrt{12-12\sqrt{3}+9}\)

\(=\sqrt{\left[1+\sqrt{3}\right]^2}-\sqrt{\left[2\sqrt{3}-3\right]^2}\)

\(=\left(1+\sqrt{3}\right)-\left(2\sqrt{3}-3\right)\)

\(=1+\sqrt{3}-2\sqrt{3}+3\)

\(=4-\sqrt{3}\)

chúc bn học tốt

a.\(\sqrt{7+4\sqrt{3}}=\sqrt{\left(\sqrt{3}+2\right)^2}=\left|\sqrt{3}+2\right|=\sqrt{3}+2\)

b.\(\sqrt{9-4\sqrt{5}}=\sqrt{\left(\sqrt{5}-2\right)^2}=\left|\sqrt{5}-2\right|=\sqrt{5}-2\)

c.\(\sqrt{14+6\sqrt{5}}=\sqrt{\left(\sqrt{5}+3\right)^2}=\left|\sqrt{5}+3\right|=\sqrt{5}+3\)

d.\(\sqrt{17-12\sqrt{2}}=\sqrt{\left(2\sqrt{2}-3\right)^2}=\left|2\sqrt{2}-3\right|=3-2\sqrt{2}\)

\(\sqrt{6-4\sqrt{2}}\)\(+\sqrt{22-12\sqrt{2}}\)

\(=\sqrt{4-4\sqrt{2}+2}\)\(+\sqrt{18-12\sqrt{2}+4}\)

\(=\sqrt{\left(2-\sqrt{2}\right)^2}\)\(+\sqrt{\left(2-3\sqrt{2}\right)^2}\)

\(=2-\sqrt{2}+3\sqrt{2}-2\)

\(=\left(2-2\right)+\left(-\sqrt{2}+3\sqrt{2}\right)\)

\(=0+2\sqrt{2}\)\(=2\sqrt{2}\)

\(\sqrt{17-12\sqrt{2}}\)\(+\sqrt{9+4\sqrt{2}}\)

\(=\sqrt{\left(3-2\sqrt{2}\right)^2}\)\(+\sqrt{\left(2\sqrt{2}+1\right)^2}\)

\(=\left|3-2\sqrt{2}\right|\)\(+\left|2\sqrt{2}+1\right|\)

\(=3-2\sqrt{2}\)\(+2\sqrt{2}+1\)

\(=\left(3+1\right)+\left(-2\sqrt{2}+2\sqrt{2}\right)\)

\(=4+0=4\)

a: Ta có: \(\sqrt{8+2\sqrt{15}}-\sqrt{6+2\sqrt{5}}\)

\(=\sqrt{5}+\sqrt{3}-\sqrt{5}-1\)

\(=\sqrt{3}-1\)

b: Ta có: \(\sqrt{17-2\sqrt{72}}+\sqrt{19+2\sqrt{18}}\)

\(=3-2\sqrt{2}+3\sqrt{2}+1\)

\(=4+\sqrt{2}\)

c: Ta có: \(\sqrt{12-2\sqrt{32}}+\sqrt{9+4\sqrt{2}}\)

\(=2\sqrt{2}-2+2\sqrt{2}+1\)

\(=4\sqrt{2}-1\)

a)

\(\sqrt{8+2\sqrt{15}}-\sqrt{6+2\sqrt{5}}\\ =\sqrt{5+2\sqrt{5}\cdot\sqrt{3}+3}-\sqrt{5+2\sqrt{5}\cdot\sqrt{1}+1}\\ =\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{1}\right)^2}\\ =\sqrt{5}+\sqrt{3}-\sqrt{5}-\sqrt{1}\\ =\sqrt{3}-\sqrt{1}\)

b)

\(\sqrt{17-2\sqrt{72}}+\sqrt{19+2\sqrt{18}}\\ =\sqrt{9-2\sqrt{9}\cdot\sqrt{8}+8}+\sqrt{18+2\sqrt{18}\cdot\sqrt{1}+1}\\ =\sqrt{\left(3-2\sqrt{2}\right)^2}+\sqrt{\left(3\sqrt{2}+1\right)^2}\\ =3-2\sqrt{2}+3\sqrt{2}+1\\ =4+\sqrt{2}\)

c)

\(\sqrt{12-2\sqrt{32}}+\sqrt{9+4\sqrt{2}}\\ =\sqrt{8-2\sqrt{8}\cdot\sqrt{4}+4}+\sqrt{8+2\sqrt{8}\cdot\sqrt{1}+1}\\ =\sqrt{\left(2\sqrt{2}-2\right)^2}+\sqrt{\left(2\sqrt{2}+1\right)^2}\\ =2\sqrt{2}-2+2\sqrt{2}+1\\ =4\sqrt{2}-1\)

Trả lời 

\(\sqrt{17+12\sqrt{2}}=\sqrt{9+12\sqrt{2}+8}\)

                                   \(=\sqrt{\left(3+2\sqrt{2}\right)^2}\)

                                   \(=3+2\sqrt{2}\)