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Question Number 139165 by mathlove last updated on 23/Apr/21

Answered by mr W last updated on 23/Apr/21

13+(√(48))=(2(√3))^2 +2×2(√3)+1=(2(√3)+1)^2 5−(√(13+(√(48))))=5−(2(√3)+1)=((√3))^2 −2(√3)+1=((√3)−1)^2 3+(√(5−(√(13+(√(48))))))=3+((√3)−1)=2+(√3) (√(2−(√3)))(√(3+(√(5−(√(13+(√(48))))))))=(√(2−(√3)))(√(2+(√3))) =(√(2^2 −((√3))^2 ))=(√(4−3))=1

$$\mathrm{13}+\sqrt{\mathrm{48}}=\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} +\mathrm{2}×\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{1}=\left(\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{5}−\sqrt{\mathrm{13}+\sqrt{\mathrm{48}}}=\mathrm{5}−\left(\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{1}\right)=\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} −\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{1}=\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{3}+\sqrt{\mathrm{5}−\sqrt{\mathrm{13}+\sqrt{\mathrm{48}}}}=\mathrm{3}+\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)=\mathrm{2}+\sqrt{\mathrm{3}} \\ $$$$\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\sqrt{\mathrm{3}+\sqrt{\mathrm{5}−\sqrt{\mathrm{13}+\sqrt{\mathrm{48}}}}}=\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$$$=\sqrt{\mathrm{2}^{\mathrm{2}} −\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }=\sqrt{\mathrm{4}−\mathrm{3}}=\mathrm{1} \\ $$

Commented by mathlove last updated on 23/Apr/21

thanks sir

$${thanks}\:\:\:{sir} \\ $$

Commented by greg_ed last updated on 23/Apr/21

cool !

$$\mathrm{cool}\:!\: \\ $$

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