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

\(=3-\sqrt{3}-2\sqrt{3}+3=6-3\sqrt{3}\)

`A=\sqrt{12-6\sqrt{3}}-\sqrt{21-12\sqrt{3}}`

`A=\sqrt{(\sqrt{3})^2-2.\sqrt{3}.2+3^2}-\sqrt{3^2-2.3.2\sqrt{3}+(2\sqrt{3})^2}`

`A=\sqrt{(\sqrt{3}-3)^2}-\sqrt{(3-2\sqrt{3})^2}`

`A=|\sqrt{3}-3|-|3-2\sqrt{3}|`

`A=(3-\sqrt{3})-(2\sqrt{3}-3)`

`A=3-\sqrt{3}-2\sqrt{2}+3`

`A=6-3\sqrt{3}`

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

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

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

= \(6\)

1. \(\sqrt{7+2\sqrt{10}}-\sqrt{7-2\sqrt{10}}=\sqrt{\left(\sqrt{5}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}\\ =\sqrt{5}+\sqrt{2}-\sqrt{5}+\sqrt{2}=2\sqrt{2}\)

2. \(\sqrt{12-6\sqrt{3}}+\sqrt{21-12\sqrt{3}}=\sqrt{\left(3-\sqrt{3}\right)^2}+\sqrt{\left(2\sqrt{3}-3\right)^2}\\ =3-\sqrt{3}+2\sqrt{3}-3=\sqrt{3}\)

3. \(\sqrt{33-12\sqrt{6}}+\sqrt{15-6\sqrt{6}}=\sqrt{\left(2\sqrt{6}-3\right)^2}+\sqrt{\left(3+\sqrt{6}\right)^2}\\ =2\sqrt{6}-3+3+\sqrt{6}=3\sqrt{6}\)

1.\(\sqrt{7+2\sqrt{10}}-\sqrt{7-2\sqrt{10}}=\sqrt{\left(\sqrt{2}+\sqrt{5}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}\)

\(=\sqrt{5}+\sqrt{2}-\left(\sqrt{5}-\sqrt{2}\right)=2\sqrt{2}\)

2. \(\sqrt{12-6\sqrt{3}+\sqrt{21-12\sqrt{3}}}=\sqrt{12-6\sqrt{3}+\sqrt{\left(3-2\sqrt{3}\right)^2}}\)

\(=\sqrt{12-6\sqrt{3}+2\sqrt{3}-3}=\sqrt{9-4\sqrt{3}}\)

3. \(\sqrt{33-12\sqrt{6}}+\sqrt{15-6\sqrt{6}}=\sqrt{\left(2\sqrt{6}-3\right)^2}+\sqrt{\left(\sqrt{6}-3\right)^2}\)

\(=2\sqrt{6}-3+3-\sqrt{6}=\sqrt{6}\)

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

\(\sqrt{8-2\sqrt{12}}=\sqrt{\left(\sqrt{6}-\sqrt{2}\right)^2}=\left|\sqrt{6}-\sqrt{2}\right|=\sqrt{6}-\sqrt{2}\)

\(\sqrt{21+6\sqrt{6}}=\sqrt{\left(3\sqrt{2}-\sqrt{3}\right)^2}=\left|3\sqrt{2}-\sqrt{3}\right|=3\sqrt{2}-\sqrt{3}\)

\(\sqrt{15-6\sqrt{6}}=\sqrt{\left(3-\sqrt{6}\right)^2}=\left|3-\sqrt{6}\right|=3-\sqrt{6}\)

\(\sqrt{29-12\sqrt{5}}=\sqrt{\left(2\sqrt{5}-3\right)^2}=\left|2\sqrt{5}-3\right|=2\sqrt{5}-3\)

\(\sqrt{41+12\sqrt{5}}=\sqrt{\left(6+\sqrt{5}\right)^2}=6+\sqrt{5}\)

\(D=\sqrt{9+6\sqrt{2}}-\sqrt{9-6\sqrt{2}}-\sqrt{21-12\sqrt{3}}\)

\(D=\sqrt{9+2.\sqrt{3}.\sqrt{3}.\sqrt{2}}-\sqrt{9-2\sqrt{3}.\sqrt{3}.\sqrt{2}}-\sqrt{21-2.2\sqrt{3}.3}\)

\(D=\sqrt{\left(\sqrt{6}\right)^2+2\sqrt{6}.\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{6}\right)^2-2\sqrt{6}.\sqrt{3}+\left(\sqrt{3}\right)^2}\)

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

\(D=\sqrt{\left(\sqrt{6}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{6}-\sqrt{3}\right)^2}-\sqrt{\left(3-2\sqrt{3}\right)^2}\)

\(D=\sqrt{6}+\sqrt{3}-\sqrt{6}+\sqrt{3}-2\sqrt{3}+3=3\)

A=\(\sqrt{12-6\sqrt{3}}+\sqrt{21-12\sqrt{3}}\)

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

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

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

a)

\(\sqrt{12}-\sqrt{27}+\sqrt{3}=\sqrt{4}.\sqrt{3}-\sqrt{9}.\sqrt{3}+\sqrt{3}=2\sqrt{3}-3\sqrt{3}+\sqrt{3}\)

\(=\sqrt{3}(2-3+1)=0\)

b)

\(\sqrt{252}-\sqrt{700}+\sqrt{1008}-\sqrt{448}=\sqrt{4}.\sqrt{63}-\sqrt{4}.\sqrt{175}+\sqrt{4}.\sqrt{252}-\sqrt{4}.\sqrt{112}\)

\(=2(\sqrt{63}-\sqrt{175}+\sqrt{252}-\sqrt{112})\)

\(=2(\sqrt{9}.\sqrt{7}-\sqrt{25}.\sqrt{7}+\sqrt{36}.\sqrt{7}-\sqrt{16}.\sqrt{7})\)

\(=2(3\sqrt{7}-5\sqrt{7}+6\sqrt{7}-4\sqrt{7})=2\sqrt{7}(3-5+6-4)=0\)

------------------

\(\sqrt{3}(\sqrt{12}+\sqrt{27}-\sqrt{3})=\sqrt{36}+\sqrt{81}-\sqrt{9}\)

\(=\sqrt{6^2}+\sqrt{9^2}-\sqrt{3^2}=6+9-3=12\)

c)

\(\frac{\sqrt{6}+\sqrt{10}}{\sqrt{21}+\sqrt{35}}=\frac{\sqrt{2}.\sqrt{3}+\sqrt{2}.\sqrt{5}}{\sqrt{7}.\sqrt{3}+\sqrt{7}.\sqrt{5}}=\frac{\sqrt{2}(\sqrt{3}+\sqrt{5})}{\sqrt{7}(\sqrt{3}+\sqrt{5})}=\frac{\sqrt{2}}{\sqrt{7}}\)

\(\frac{\sqrt{405}+3\sqrt{27}}{3\sqrt{3}+\sqrt{45}}=\frac{\sqrt{81}.\sqrt{5}+3\sqrt{9}.\sqrt{3}}{3\sqrt{3}+\sqrt{9}.\sqrt{5}}=\frac{9\sqrt{5}+9\sqrt{3}}{3\sqrt{3}+3\sqrt{5}}\)

\(=\frac{3(3\sqrt{5}+3\sqrt{3})}{3\sqrt{3}+3\sqrt{5}}=3\)

d)

\(\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}-\sqrt{6}-\sqrt{9}-\sqrt{12}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}-(\sqrt{6}+\sqrt{9}+\sqrt{12})}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}-(\sqrt{2}.\sqrt{3}+\sqrt{3}.\sqrt{3}+\sqrt{3}.\sqrt{4})}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}-\sqrt{3}(\sqrt{2}+\sqrt{3}+\sqrt{4})}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\frac{(\sqrt{2}+\sqrt{3}+\sqrt{4})(1-\sqrt{3})}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=1-\sqrt{3}\)

a) \(\sqrt{19-6\sqrt{2}}=3\sqrt{2}-1\)

b) \(\sqrt{11-6\sqrt{2}}=3-\sqrt{2}\)

d) \(\sqrt{21+12\sqrt{3}}=2\sqrt{3}+3\)

e) \(\sqrt{57-40\sqrt{2}}=4\sqrt{2}-5\)