What is the conjugate of `sqrt((a+b))+7` ?

It actually does not matter. You can use `-sqrt((a+b))+7` as the conjugate, or `sqrt((a+b))-7`.

By convention, if a binomial expression contains a rational and an irrational term, then we usually put the rational term first, e.g.:

`2+3sqrt(5)`

Then the radical conjugate is usually taken to be:

`2-3sqrt(5)`

i.e. reversing the sign of the irrational term.

Note that that does not cover the case of:

`sqrt(2)+sqrt(3)`

for which we could use the following expression as a conjugate:

`sqrt(2)-sqrt(3)`

Alternatively, we could use the following expression as a conjugate:

`sqrt(3)-sqrt(2)`

The fundamental idea is that a conjugate is an expression which when multiplied by the original results in a rational result.

For binomials with terms that involve rationals and square roots (but not square roots of square roots), you can form a conjugate by reversing the sign of either term.

This works because of the difference of squares identity:

`A^2-B^2 = (A-B)(A+B)`

So by reversing the sign of one term, we end up with a product only involving squares of the original terms.