Question

Объясните принцип решения примера, пожалуйста. $\frac{53}{8- \sqrt{11} } + \frac{2}{ \sqrt{13}+ \sqrt{11} } - \frac{9}{ \sqrt{13} +2}$

Answer 1

$1) \ \frac{53}{8-\sqrt{11}}+\frac{2}{\sqrt{13}+\sqrt{11}}= \frac{53( \sqrt{13}+ \sqrt{11})+2(8- \sqrt{11}) }{(8-\sqrt{11})(\sqrt{13}+\sqrt{11})}= \\\ = \frac{53 \sqrt{13}+ 53\sqrt{11}+16-2 \sqrt{11}}{8 \sqrt{13}+8 \sqrt{11}- \sqrt{143}-11 }= \frac{53 \sqrt{13}+ 51\sqrt{11}+16}{8 \sqrt{13}+8 \sqrt{11}- \sqrt{143}-11 }$

$2) \ \frac{53 \sqrt{13}+ 51\sqrt{11}+16}{8 \sqrt{13}+8 \sqrt{11}- \sqrt{143}-11 } - \frac{9}{ \sqrt{13} +2} = \\\ =\frac{(\sqrt{13} +2)(53 \sqrt{13}+ 51\sqrt{11}+16)-9(8 \sqrt{13}+8 \sqrt{11}- \sqrt{143}-11)}{(8 \sqrt{13}+8 \sqrt{11}- \sqrt{143}-11)(\sqrt{13} +2) }= \\\ =\frac{53\cdot13+51 \sqrt{143}+16 \sqrt{13}+106 \sqrt{13}+102 \sqrt{11}+32-72 \sqrt{13}-72 \sqrt{11}+9 \sqrt{143}+99 }{8\cdot13+8 \sqrt{143}- \sqrt{13\cdot11\cdot13}-11 \sqrt{13} +16 \sqrt{13}+16 \sqrt{11}- 2\sqrt{143}-22 }=$
$=\frac{689+60 \sqrt{143}+50 \sqrt{13}+30 \sqrt{11}+131 }{104+6 \sqrt{143}- 13\sqrt{11}+5 \sqrt{13}+16 \sqrt{11}-22 }= \frac{820+60 \sqrt{143}+50 \sqrt{13}+30 \sqrt{11} }{82+6 \sqrt{143}+5 \sqrt{13}+3 \sqrt{11} }= \\\ =\frac{10(82+6 \sqrt{143}+5 \sqrt{13}+3 \sqrt{11}) }{82+6 \sqrt{143}+5 \sqrt{13}+3 \sqrt{11} }=10$