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

Jan 29, 2017

You can use $- \sqrt{\left(a + b\right)} + 7$ or $\sqrt{\left(a + b\right)} - 7$ as the conjugate of $\sqrt{\left(a + b\right)} + 7$

#### Explanation:

It actually does not matter. You can use $- \sqrt{\left(a + b\right)} + 7$ as the conjugate, or $\sqrt{\left(a + b\right)} - 7$.

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

$2 + 3 \sqrt{5}$

Then the radical conjugate is usually taken to be:

$2 - 3 \sqrt{5}$

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:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

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