Question

What is radical conjugate?

Answer 1

Hint: Assuming that this is a maths question rather than a chemistry question, the radical conjugate of \[a + \sqrt[b]{c}\] is \[a - \sqrt[b]{c}\]
For \[a + b\], it's \[a - b\]
 A conjugate is the same two term expression but with the sign in the middle changed. So, for example, if you have\[\left( {3 + \sqrt 2 } \right)\] , the conjugate would be \[\left( {3 - \sqrt 2 } \right)\] .

Complete step by step solution:
A conjugate is a binomial where the second term is the opposite of the second term of another binomial. In other words, if I have the expression \[a + b\], the conjugate of my expression is \[a - b\]. If I were to add the second terms of these two expressions, I would get 0.
Conjugates are relative to one another: two numbers can be conjugates of each other, but one number cannot simply be a conjugate in a vacuum. Conjugation also only happens between binomials: there must be exactly two terms in an equation for it to have a conjugate.
When simplifying a rational expression such as:
\[\dfrac{{1 + \sqrt 3 }}{{2 + \sqrt 3 }}\]
we want to rationalize the denominator \[\left( {2 + \sqrt 3 } \right)\] by multiplying by the radical conjugate \[\left( {2 - \sqrt 3 } \right)\], formed by inverting the sign on the radical (square root) term.
\[\dfrac{{1 + \sqrt 3 }}{{2 + \sqrt 3 }} = \dfrac{{1 + \sqrt 3 }}{{2 + \sqrt 3 }} \times \dfrac{{2 - \sqrt 3 }}{{2 - \sqrt 3 }} = \dfrac{{ - 1 + \sqrt 3 }}{{4 - 3}} = \sqrt 3 - 1\]

Additional Information:
The most significant property of conjugates is that if you multiply them together, the result will always be of the form:
\[(a + b)(a - b) = ({a^2} - {b^2})\]
Similarly, the roots of an equation of the form \[{a^2} - {b^2}\] will always be a pair of conjugates.
A complex conjugate is actually a special case of the radical conjugate in which the radical is \[i = \sqrt { - 1} \]

Note:
The property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator.
The pair of factors, with the second factor differing only in the one sign in the middle, is very important; in fact, this "same except for the sign in the middle" second factor has its own name.