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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