Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 439

а) a = 55; b = 216
$$ \sqrt{55+\sqrt{216}}=\sqrt{\frac{55+\sqrt{55^2-216}}{2}}+\sqrt{\frac{55-\sqrt{55^2-216}}{2}}= $$ $$ \sqrt{\frac{55+\sqrt{3025-216}}{2}}+\sqrt{\frac{55-\sqrt{3025-216}}{2}}= $$ $$ \sqrt{\frac{55+\sqrt{2809}}{2}}+\sqrt{\frac{55-\sqrt{2809}}{2}}=\sqrt{\frac{55+53}{2}}+\sqrt{\frac{55-53}{2}}=\sqrt{\frac{108}{2}}+\sqrt{\frac{2}{2}} $$ = $$\sqrt{54}$$ + $$\sqrt{1}$$ = $$\sqrt{54}$$ + 1
б) a = 86; b = 5460
$$ \sqrt{86-\sqrt{5460}}=\sqrt{\frac{86+\sqrt{86^2-5460}}{2}}-\sqrt{\frac{86-\sqrt{86^2-5460}}{2}}= $$ $$ \sqrt{\frac{86+\sqrt{7396-5460}}{2}}-\sqrt{\frac{86-\sqrt{7396-5460}}{2}}= $$ $$ \sqrt{\frac{86+\sqrt{1936}}{2}}-\sqrt{\frac{86-\sqrt{1936}}{2}}=\sqrt{\frac{86+44}{2}}-\sqrt{\frac{86-44}{2}}=\sqrt{\frac{130}{2}}-\sqrt{\frac{42}{2}}= $$ = $$\sqrt{65}$$ − $$\sqrt{21}$$
в) a = 17; b = 288
$$ \sqrt{17+\sqrt{288}}=\sqrt{\frac{17+\sqrt{17^2-288}}{2}}+\sqrt{\frac{17-\sqrt{17^2-288}}{2}}= $$ $$ \sqrt{\frac{17+\sqrt{289-288}}{2}}+\sqrt{\frac{17-\sqrt{289-288}}{2}}= $$ $$ \sqrt{\frac{17+\sqrt{1}}{2}}+\sqrt{\frac{17-\sqrt{1}}{2}}=\sqrt{\frac{17+1}{2}}+\sqrt{\frac{17-1}{2}}=\sqrt{\frac{18}{2}}+\sqrt{\frac{16}{2}}= $$ = $$\sqrt{9}$$ + $$\sqrt{8}$$ = 3 + $$\sqrt{8}$$
г) a = 32; b = 1008
$$ \sqrt{32-\sqrt{1008}}=\sqrt{\frac{32+\sqrt{32^2-1008}}{2}}-\sqrt{\frac{32-\sqrt{32^2-1008}}{2}}= $$ $$ =\sqrt{\frac{32+\sqrt{1024-1008}}{2}}-\sqrt{\frac{32-\sqrt{1024-1008}}{2}}= $$ $$ \sqrt{\frac{32+\sqrt{16}}{2}}-\sqrt{\frac{32-\sqrt{16}}{2}}=\sqrt{\frac{32+4}{2}}-\sqrt{\frac{32-4}{2}}=\sqrt{\frac{36}{2}}-\sqrt{\frac{28}{2}}= $$ = $$\sqrt{18}$$ − $$\sqrt{14}$$