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Question-211659




Question Number 211659 by efronzo1 last updated on 15/Sep/24
Commented by Frix last updated on 15/Sep/24
Simply transform it ⇒ n=3
$$\mathrm{Simply}\:\mathrm{transform}\:\mathrm{it}\:\Rightarrow\:{n}=\mathrm{3} \\ $$
Answered by BHOOPENDRA last updated on 15/Sep/24
(√5) +n(√3)−2(√2)=((√5)+(√2))((√(15))+3−(√(10))−(√6))  (√5)+n(√3)−2(√2)=3(√3)−2(√2)+(√5)  compare both side   n=3
$$\sqrt{\mathrm{5}}\:+{n}\sqrt{\mathrm{3}}−\mathrm{2}\sqrt{\mathrm{2}}=\left(\sqrt{\mathrm{5}}+\sqrt{\mathrm{2}}\right)\left(\sqrt{\mathrm{15}}+\mathrm{3}−\sqrt{\mathrm{10}}−\sqrt{\mathrm{6}}\right) \\ $$$$\sqrt{\mathrm{5}}+{n}\sqrt{\mathrm{3}}−\mathrm{2}\sqrt{\mathrm{2}}=\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{2}\sqrt{\mathrm{2}}+\sqrt{\mathrm{5}} \\ $$$${compare}\:{both}\:{side}\: \\ $$$${n}=\mathrm{3} \\ $$
Answered by a.lgnaoui last updated on 15/Sep/24
(√5) +n(√3) −2(√2) =((√5) +(√3))((√5) +(√2) )(.(√3) −(√2) )  =[5+(√(10)) +(√(15)) +(√6) )]((√3) −(√2) )=  =5(√3) −5(√2) +(√(30)) −2(√5) +3(√5) −(√(30))     +3(√2) −2(√3)   =3(√3) +(√5) −2(√2)     ⇒(√5) +n(√3) −2(√2) =(√5) +3(√3) −2(√2)                      n=3
$$\sqrt{\mathrm{5}}\:+\boldsymbol{\mathrm{n}}\sqrt{\mathrm{3}}\:−\mathrm{2}\sqrt{\mathrm{2}}\:=\left(\sqrt{\mathrm{5}}\:+\sqrt{\mathrm{3}}\right)\left(\sqrt{\mathrm{5}}\:+\sqrt{\mathrm{2}}\:\right)\left(.\sqrt{\mathrm{3}}\:−\sqrt{\mathrm{2}}\:\right) \\ $$$$\left.=\left[\mathrm{5}+\sqrt{\mathrm{10}}\:+\sqrt{\mathrm{15}}\:+\sqrt{\mathrm{6}}\:\right)\right]\left(\sqrt{\mathrm{3}}\:−\sqrt{\mathrm{2}}\:\right)= \\ $$$$=\mathrm{5}\sqrt{\mathrm{3}}\:−\mathrm{5}\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{30}}\:−\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{3}\sqrt{\mathrm{5}}\:−\sqrt{\mathrm{30}}\: \\ $$$$\:\:+\mathrm{3}\sqrt{\mathrm{2}}\:−\mathrm{2}\sqrt{\mathrm{3}}\: \\ $$$$=\mathrm{3}\sqrt{\mathrm{3}}\:+\sqrt{\mathrm{5}}\:−\mathrm{2}\sqrt{\mathrm{2}}\: \\ $$$$ \\ $$$$\Rightarrow\sqrt{\mathrm{5}}\:+\mathrm{n}\sqrt{\mathrm{3}}\:−\mathrm{2}\sqrt{\mathrm{2}}\:=\sqrt{\mathrm{5}}\:+\mathrm{3}\sqrt{\mathrm{3}}\:−\mathrm{2}\sqrt{\mathrm{2}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{n}}=\mathrm{3} \\ $$

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