`a)x=sqrt{7-4sqrt3}+sqrt{4-2sqrt3}`
`=sqrt{4-2.2sqrt3+3}+sqrt{3-sqrt3+1}`
`=sqrt{(2-sqrt3)^2}+\sqrt{(sqrt3-1)^2}`
`=2-sqrt3+sqrt3-1=1`
`b)x=sqrt{61+28sqrt3}-sqrt{21+12sqrt3}`
`=sqrt{49+2.7.2sqrt3+12}-sqrt{12+2.2sqrt3.3+9}`
`=sqrt{(7+2sqrt3)^2}-sqrt{(2sqrt3+3)^2}`
`=7+2sqrt3-2sqrt3-3`
`=4`
`c)x=sqrt{5+(1-8sqrt5)/16}+sqrt{45/4-5sqrt5}`
`=sqrt{(80+1-8sqrt5)/16}+sqrt{(45-20sqrt5)/4}`
`=sqrt{(4sqrt5-1)^2/16}+sqrt{(25-2.5.2sqrt5+20)/4}`
`=(4sqrt5-1)/4+sqrt{(5-2sqrt5)/4}`
`=(4sqrt5-1)/4+(5-2sqrt5)/2`
`=(4sqrt5-1+10-4sqrt5)/4=9/4`
`d)x=sqrt{38+12sqrt2}-sqrt{11-6sqrt2}`
`=sqrt{36+2.6.sqrt2+2}-sqrt{9-2.3.sqrt2+2}`
`=sqrt{(6+sqrt2)^2}-sqrt{(3-sqrt2)^2}`
`=6+sqrt2-3+sqrt2=3+2sqrt2`
a) \(x=\sqrt{7-4\sqrt{3}}+\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{2^2-2.2.\sqrt{3}+\left(\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}\right)^2-2.1.\sqrt{3}+1^2}\)
\(=\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}=\left|2-\sqrt{3}\right|+\left|\sqrt{3}-1\right|\)
\(=2-\sqrt{3}+\sqrt{3}-1=1\)
b) \(x=\sqrt{61+28\sqrt{3}}-\sqrt{21+12\sqrt{3}}\)
\(=\sqrt{7^2+2.7.2\sqrt{3}+\left(2\sqrt{3}\right)^2}-\sqrt{3^2+2.3.2\sqrt{3}+\left(2\sqrt{3}\right)^2}\)
\(=\sqrt{\left(7+2\sqrt{3}\right)^2}-\sqrt{\left(3+2\sqrt{3}\right)^2}=\left|7+2\sqrt{3}\right|-\left|3+2\sqrt{3}\right|\)
\(=7+2\sqrt{3}-3-2\sqrt{3}=4\)
c) \(x=\sqrt{5+\dfrac{1-8\sqrt{5}}{16}}+\sqrt{\dfrac{45}{4}-5\sqrt{5}}\text{}\text{}=\sqrt{\dfrac{81-8\sqrt{5}}{16}}+\sqrt{\dfrac{45-20\sqrt{5}}{4}}\)
\(=\sqrt{\dfrac{\left(4\sqrt{5}\right)^2-2.4\sqrt{5}.1+1^2}{4^2}}+\sqrt{\dfrac{5^2-2.5.2\sqrt{5}+\left(2\sqrt{5}\right)^2}{2^2}}\)
\(=\sqrt{\dfrac{\left(4\sqrt{5}-1\right)^2}{4^2}}+\sqrt{\dfrac{\left(5-2\sqrt{5}\right)^2}{2^2}}=\left|\dfrac{4\sqrt{5}-1}{4}\right|+\left|\dfrac{5-2\sqrt{5}}{2}\right|\)
\(=\dfrac{4\sqrt{5}-1}{4}+\dfrac{5-2\sqrt{5}}{2}=\dfrac{4\sqrt{5}-1+10-4\sqrt{5}}{4}=\dfrac{9}{4}\)
d) \(x=\sqrt{38+12\sqrt{2}}-\sqrt{11-6\sqrt{2}}\)
\(=\sqrt{6^2+2.6.\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{3^2-2.3.\sqrt{2}+\left(\sqrt{2}\right)^2}\)
\(=\sqrt{\left(6+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}=\left|6+\sqrt{2}\right|-\left|3-\sqrt{2}\right|\)
\(=6+\sqrt{2}-3+\sqrt{2}=3+2\sqrt{2}\)
