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Question Number 85864 by john santu last updated on 26/Mar/20

simplify the expression  (√(6+2(√(8(√3)−10)))) − (√(7−(√3)))  in  the form (√((√a)+b)) ?

$${simplify}\:{the}\:{expression} \\ $$$$\sqrt{\mathrm{6}+\mathrm{2}\sqrt{\mathrm{8}\sqrt{\mathrm{3}}−\mathrm{10}}}\:−\:\sqrt{\mathrm{7}−\sqrt{\mathrm{3}}}\:\:{in} \\ $$$${the}\:{form}\:\sqrt{\sqrt{{a}}+{b}}\:? \\ $$

Answered by jagoll last updated on 26/Mar/20

by trick !   8(√3) −10 = 7(√3) +(√3) −7−3  = 7((√3)−1) −(√3) ((√3) −1)  = (7−(√3) )((√3) −1)  let (√(7−(√3) )) = x and (√((√3)−1)) =y  x^2  + y^2  = 7−(√3) +(√3) −1 = 6  hence a = 3 & b = −1

$$\mathrm{by}\:\mathrm{trick}\:!\: \\ $$$$\mathrm{8}\sqrt{\mathrm{3}}\:−\mathrm{10}\:=\:\mathrm{7}\sqrt{\mathrm{3}}\:+\sqrt{\mathrm{3}}\:−\mathrm{7}−\mathrm{3} \\ $$$$=\:\mathrm{7}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)\:−\sqrt{\mathrm{3}}\:\left(\sqrt{\mathrm{3}}\:−\mathrm{1}\right) \\ $$$$=\:\left(\mathrm{7}−\sqrt{\mathrm{3}}\:\right)\left(\sqrt{\mathrm{3}}\:−\mathrm{1}\right) \\ $$$$\mathrm{let}\:\sqrt{\mathrm{7}−\sqrt{\mathrm{3}}\:}\:=\:\mathrm{x}\:\mathrm{and}\:\sqrt{\sqrt{\mathrm{3}}−\mathrm{1}}\:=\mathrm{y} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{7}−\sqrt{\mathrm{3}}\:+\sqrt{\mathrm{3}}\:−\mathrm{1}\:=\:\mathrm{6} \\ $$$$\mathrm{hence}\:\mathrm{a}\:=\:\mathrm{3}\:\&\:\mathrm{b}\:=\:−\mathrm{1} \\ $$

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