Question

How do you simplify `(4+sqrt2 )div (8 - sqrt2)`?

Answer 1

`(17+6sqrt2)/31`

Explanation:

Let's rewrite this in the fractional form:
`(4+sqrt2)/(8-sqrt2)`

Then, I would multiply, top and bottom, by `(8+sqrt2)` so that:
`(4+sqrt2)/(8-sqrt2) * (8+sqrt2)/(8+sqrt2)`

(Note that `(8+sqrt2)/(8+sqrt2)=1`, so multiplying by this factor does not change the results at all).

I chose `(8+sqrt2)` because
`(a-b)*(a+b)=a^2-b^2`
and I am hoping it will make the solution easier.

Then the fraction becomes:
`((4+sqrt2)(8+sqrt2))/(8^2 - (sqrt2)^2) = (32+12sqrt2+2)/(64-2)`
which is
`=(34+12sqrt2)/(62)`
We can factor out 2 from the top and bottom:
`=(2*(17+6sqrt2))/(2*31)`
`=(17+6sqrt2)/31`