# calculate E=(a^(-1)+b^(-1))/a^(-1)-b^(-1) if: a=((5√3+√50)(5-√24))/√75-5√2 b=√(7+4√3)+√(7-4√3)

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You need to evaluate `a^(-1)` and `b^(-1)` such that:

`a^(-1) = 1/a, b^(-1) = 1/b`

`a^(-1) = (sqrt 75 - 5sqrt 2)/((5sqrt 3 + sqrt 50)(5 - sqrt 24))`

`a^(-1) = (sqrt 75 - 5sqrt 2)/(25sqrt 3 - 30sqrt 2 + 25sqrt 2 - 20sqrt3)`

`a^(-1) = (5sqrt 3 - 5sqrt 2)/(5sqrt 3 - 5sqrt 2)`

Reducing duplicate factors yields:

`a^(-1) = 1`

You need to evaluate `b^(-1)` such that:

`b^(-1) = 1/(sqrt(7+4sqrt3) + sqrt(7-4sqrt3))`

`b^(-1) = (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(7+4sqrt3-7+4sqrt3)`

`b^(-1) = (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3)`

Evaluating E yields:

`E = (1 + (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3))/((sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3))`

Reducing duplicate factors yields:

`E = (8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(sqrt(7+4sqrt3) - sqrt(7-4sqrt3))`

`E = ((8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))(sqrt(7+4sqrt3) + sqrt(7-4sqrt3)))/(8sqrt 3)`

Hence, evaluating the given expression yields `E = ((8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))(sqrt(7+4sqrt3) + sqrt(7-4sqrt3)))/(8sqrt 3).`

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First wirte  `(a^(-1)+b^(-1))/(a^(-1)-b^(-1))`  as   `(1/a+1/b)/(1/a-1/b)=` `((b+a)/(ab))/((b-a)/(ab))` `=(b+a)/(b-a)`

Now   `a=((sqrt(3)+sqrt(2))(5-2sqrt(6)))/(sqrt(3)-sqrt(2))`

multypling  for  `(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))`  we get:

`a=(5+2sqrt(6))(5-2sqrt(6))=1`

`b= sqrt(7+4sqrt(3))+sqrt(7-4sqrt(3))`

`b= sqrt(7+sqrt(48))+sqrt(7-sqrt(48))`

`b=2sqrt((7+sqrt(49-48))/2)` `=+-4`

So E has two values:

`E_1=(4+1)/(4-1)=5/3`

`E_2=(-4+1)/(-4-1)=3/5`