Question

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

Answer 1

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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