Question

(5√3-3√5):(5√3+3√5)=?
60:(3√2+2√3-√30)=?

Answer 1

(5√3-3√5):(5√3+3√5)=
5√3-3√5/5√3+3√5
(5√3-3√5)(5√3-3√5)/30
(5√3-3√5)²/30
75-30√15+45/30
120-30√15/30
30(4-√15)/30=4-√15

60:(3√2+2√3-√30)=
60/3√2+2√3-√30
60(3√2+2√3-√30)/12√6
5(3√2+2√3-√30)/√6
15√2+10√3+5√30/√6
(15√3+10√3+5√30)√6/6
15√12+10√18+5√180/6
30√3+30√2+30√5/6
6(30√3+30√2+30√5)/6
6(5√3+5√2+5√5)/6
=5√3+5√2+5√5

Answer 2

$\frac{5\sqrt{3}-3\sqrt{5}}{5\sqrt{3}+3\sqrt{5}}=\frac{(5\sqrt{3}-3\sqrt{5})^{2}}{(5\sqrt{3})^{2}-(3\sqrt{5})^{2}}=$

$=\frac{75-30\sqrt{15}+45}{75-45}=$

$=\frac{120-30\sqrt{15}}{30}=$

$=\frac{30(4-\sqrt{15})}{30}=$

$\boxed{=4-\sqrt{15}}$


60:(3√2+2√3-√30)=
=60 :3√2 +60:2√3 -60:√30=

$=\frac{60}{3\sqrt{2}}+\frac{60}{2\sqrt{3}}-\frac{60}{\sqrt{30}}=$

$=\frac{60\sqrt{2}}{6}+\frac{60\sqrt{3}}{6}-\frac{60\sqrt{30}}{30}=$

$\boxed{=10\sqrt{2}+10\sqrt{3}-2\sqrt{30}}$