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解?$：(1)$?原式?$=3 \sqrt {3}+\frac {\sqrt {3}}{3} -(2 \sqrt {3}-\frac {\sqrt {5}}{5}+3 \sqrt {5}) $?
?$=\frac {10 \sqrt {3}}{3}-2 \sqrt {3}+\frac {14 \sqrt {5}}{5} $?
?$=\frac {10 \sqrt {3}}{3}-2 \sqrt {3}-\frac {14 \sqrt {5}}{5} $?
?$=\frac {4 \sqrt {3}}{3}-\frac {14 \sqrt {5}}{5}$?
?$(2)2 \sqrt {125}-\sqrt {28}+\sqrt {\frac {1}{20}}-\frac {1}{3} \sqrt {175} $?
?$=10 \sqrt {5}-2 \sqrt {7}+\frac {\sqrt {5}}{10}-\frac {5 \sqrt {7}}{3} $?
?$=\frac {101 \sqrt {5}}{10}-\frac {11 \sqrt {7}}{3}$?
?$\text { (3)原式 }=2 \sqrt{a-b}+(a-b) \sqrt{a-b}-a \sqrt{a-b} $?
?$=(2-b) \sqrt {a-b}$?

解?$：\frac {2}{5} \sqrt {25\ \mathrm {a}}+9 \sqrt {\frac {a}{3}}-2\ \mathrm {a}^2 \sqrt {\frac {1}{a^3}}=18 $?
?$\frac {2}{5} \sqrt {5^2 ·a}+9 \sqrt {\frac {3\ \mathrm {a}}{3^2}}-2\ \mathrm {a}^2\sqrt {\frac {a}{(a^2)^2}} =18 $?
?$\frac {2}{5} ×5 \sqrt {a}+9 ×\frac {1}{3} \sqrt {3\ \mathrm {a}}-2\ \mathrm {a}^2 ·\frac {1}{a^2} \sqrt {a}=18 $?
?$2 \sqrt {a}+3 \sqrt {3\ \mathrm {a}}-2 \sqrt {a}=18 $?
?$3 \sqrt {3\ \mathrm {a}}=18 $?
?$\sqrt {3\ \mathrm {a}}=6 $?
?$3\ \mathrm {a}=6^2 $?
?$a=12$?

解:∵?$x+\frac {1}{x}=\sqrt {8} $?
∴?$(x+\frac {1}{x})^2=x^2+2+\frac {1}{x^2}=8 $?
∴?$x^2+\frac {1}{x^2}=6 $?
∴?$x^2+\frac {1}{x^2}-2=4 $?
?$\text { 即 }(x-\frac{1}{x})^{2}=4 $?
∴?$x-\frac{1}{x}=2 \text { 或 }-2$?

解:∵?$x=2-\sqrt {3}， y=2+\sqrt {3}， $?
∴?$x^2+x y+y^2 $?
?$=x^2+2 x y+y^2-x y $?
?$=(x+y)^2-x y $?
?$=(2-\sqrt {3}+2+\sqrt {3})^2 $?
?$-(2-\sqrt {3})(2+\sqrt {3}) $?
?$=16-4+3 $?
?$=15 .$?

解?$：(1)x^2=24 $?
?$x= \pm \sqrt {24} $?
?$x_1=2 \sqrt {6}， x_2=-2 \sqrt {6}$?
?$\text { (2) } 3 \sqrt{2} x=-\sqrt{8} $?
?$x=-\sqrt {8} ·\frac {1}{3 \sqrt {2}} $?
?$x=-\frac {2}{3}$?

解?$：(1)$?原式?$=5-3=2；$?
?$(2)$?原式?$=16+3+8\sqrt 3-16-3+8\sqrt 3=16\sqrt 3；$?
?$(3)$?原式?$=\frac 12\sqrt 3-2\sqrt 6-2\sqrt 3=-\frac 32\sqrt 3-2\sqrt 6；$?
?$(4)$?原式?$=(1+x)^2-x=1+x+x^2$?

?$\text { 解: } $?∵?${x}=\sqrt {5}-2， $?
∴?$x^2=(\sqrt {5}-2)^2=5-4 \sqrt {5}+4=9-4 \sqrt {5}， $?
∴?$(9+4 \sqrt {5}) x^2-(\sqrt {5}+2) {x}+4$?
?$=(9+4 \sqrt {5})(9-4 \sqrt {5})-( \sqrt {5}+2)(\sqrt {5}-2)+4 $?
?$=81-80-(5-4)+4 $?
?$=1-1+4 $?
?$=4 .$?

解:∵?$a-b=\sqrt {3}+\sqrt {2}， b-c=\sqrt {3}-\sqrt {2}， $?
∴?$a-c=2 \sqrt {3}， $?
∴?$a^2+b^2+c^2-a b-b c-a c $?
?$=\frac {1}{2}[(a-b)^2+(b-c)^2+(a-c)^2] $?
?$=\frac {1}{2}[(\sqrt {3}+\sqrt {2})^2+(\sqrt {3}-\sqrt {2})^2. .+(2 \sqrt {3})^2] $?
?$=\frac {1}{2}(5+2 \sqrt {6}+5-2 \sqrt {6}+12) $?
?$=\frac {1}{2} ×22 $?
?$=11 .$?
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