What is the conjugate of #sqrt((a+b))+7# ?
You can use
It actually does not matter. You can use
By convention, if a binomial expression contains a rational and an irrational term, then we usually put the rational term first, e.g.:
Then the radical conjugate is usually taken to be:
i.e. reversing the sign of the irrational term.
Note that that does not cover the case of:
for which we could use the following expression as a conjugate:
Alternatively, we could use the following expression as a conjugate:
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.