Question

Multiply: √2+√5+√7. by. √2+√5-√7​

Answer 1

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$\sf\:(\sqrt{2} +\sqrt{5}+\sqrt{7})\times(\sqrt{2}+\sqrt{5}-\sqrt{7})$

$\sf↦\:\sqrt{2}(\sqrt{2}+\sqrt{5}-\sqrt{7})+\sqrt{5}(\sqrt{2}+\sqrt{5}-\sqrt{7})+\sqrt{7}(\sqrt{2}+\sqrt{5}-\sqrt{7})$

$\sf↦\:\sqrt{4}+\sqrt{10}-\sqrt{14}+(\sqrt{10}+\sqrt{25}-\sqrt{35})+(\sqrt{14}+\sqrt{35}-\sqrt{49})$

$\sf↦\:\sqrt{4}+\sqrt{10}-\sqrt{14}+\sqrt{10}+\sqrt{25}-\sqrt{35}+\sqrt{14}+\sqrt{35}-\sqrt{49}$

$\sf↦\:2+\sqrt{10}-\sqrt{14}+\sqrt{10}+5-\sqrt{35}+\sqrt{14}+\sqrt{35}-7$

$\sf↦\:2+5-7+\sqrt{10}+\sqrt{10}-\sqrt{14}+\sqrt{14}-\sqrt{35}+\sqrt{35}$

$\sf↦\:7-7+\sqrt{10}+\sqrt{10}-\sqrt{14}+\sqrt{14}-\sqrt{35}+\sqrt{35}$

$\small{\underline{\boxed{\mathrm\pink{↦\:0}}}}$

Since, everything gets cancelled.

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