Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 +
\sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $
The question is to simplify $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 +
\sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 −
\sqrt7)$ without using a calculator .
My friend has given me this challenge . I solved it by expanding
$$(a+b+c)(-a+b+c)(a-b+c)(a+b-c) = -a^4+2a^2b^2+2a^2c^2-b^4+2b^2c^2-c^4$$
and then substituting $a,b,c=\sqrt 5 , \sqrt6 , \sqrt7$ respectively to
get the answer $104$ .
But I suppose there is a more elegant and easy way to solve this problem .
Can anyone find it ?
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