How do you simplify $(sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4$ ?

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Answer 1 · 2015-02-07T14:59:15

Okay i should thank you for such a question

OK lets start

Call $sqrt 7 =sqrta and sqrt 5 =sqrtb$

So lets rewrite you equation as ${( sqrta + sqrtb) ^2}^2 +{ ( sqrta -sqrt b )^2 }^2$

So lets take the first part and simplify

$( sqrta + sqrtb) ^2 = a + 2 sqrt(ab) + b$

So lets substitute $7 + 5 + 2 sqrt35 = 12 + sqrt 35$

Square this gain and you will get that ${( sqrta + sqrtb) ^2}^2 =144 + 35 + 24 sqrt(35)= 179+ 24sqrt 35$

Similarly repeat the above steps for ${ ( sqrta -sqrt b )^2 }^2$

After substituting you should get that the above equation is $= 179 - 24sqrt35$

So finally lets put the puzzle together

$hence {( sqrta + sqrtb) ^2}^2 +{ ( sqrta -sqrt b )^2 }^2= 179 + 24 sqrt35 + 179 - 24sqrt35$

$(sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4 = 358$

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How do you simplify (sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4(sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4?