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Question Number 183041 by Shrinava last updated on 18/Dec/22

Find: (√(12−2(√(35)))) − (√(10−2(√(21)))) = ?

$$\mathrm{Find}: \\ $$$$\sqrt{\mathrm{12}−\mathrm{2}\sqrt{\mathrm{35}}}\:\:−\:\:\sqrt{\mathrm{10}−\mathrm{2}\sqrt{\mathrm{21}}}\:\:=\:\:? \\ $$

Answered by HeferH last updated on 18/Dec/22

(√(12−2(√(35)))) = (√(((√7))^2 + ((√5))^2 −2(√7)∙(√5))) ⇒ (√(((√7) − (√5))^2 )) = (√7) − (√5) (√(10−2(√(21)))) = (√(((√7))^2 + ((√3))^2 − 2(√7)∙(√3))) ⇒ (√(((√7) − (√3))^2 )) = (√7) − (√3) (√7) − (√5)−((√7) −(√3)) = (√3) − (√5)

$$\:\sqrt{\mathrm{12}−\mathrm{2}\sqrt{\mathrm{35}}}\:=\:\sqrt{\left(\sqrt{\mathrm{7}}\right)^{\mathrm{2}} \:+\:\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2}} −\mathrm{2}\sqrt{\mathrm{7}}\centerdot\sqrt{\mathrm{5}}}\:\: \\ $$$$\:\Rightarrow\:\sqrt{\left(\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }\:=\:\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{5}} \\ $$$$\:\sqrt{\mathrm{10}−\mathrm{2}\sqrt{\mathrm{21}}}\:=\:\sqrt{\left(\sqrt{\mathrm{7}}\right)^{\mathrm{2}} \:+\:\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:−\:\mathrm{2}\sqrt{\mathrm{7}}\centerdot\sqrt{\mathrm{3}}}\: \\ $$$$\:\Rightarrow\:\sqrt{\left(\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }\:=\:\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{3}} \\ $$$$\:\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{5}}−\left(\sqrt{\mathrm{7}}\:−\sqrt{\mathrm{3}}\right)\:=\:\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{5}} \\ $$

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