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# What is the conjugate of the square root of $2 +$ the square root of $3 +$ the square root of $5$?

Last updated date: 14th Jul 2024
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They are mostly irrational numbers. A conjugate is the opposite sign expression of the two values like a and b in the form of $a + b$ then the conjugation of this expression is $a - b$.
Given question is $\sqrt 2 + \sqrt 3 + \sqrt 5$ since here three are total three terms; so, then we cannot have one conjugation to find the answer, thus we use the method of conjugation for two variables step by step and we combine all in later. First, take the terms $\sqrt 2 + \sqrt 3$ and here the conjugation of the terms is $\sqrt 2 - \sqrt 3$ (the positive value is inverse is negative), similarly take the next two terms which are $\sqrt 3 + \sqrt 5$and the conjugation of the term is $\sqrt 3 - \sqrt 5$ and also the other terms becomes $\sqrt 2 + \sqrt 5$ as $\sqrt 2 - \sqrt 5$.
The first conjugation of $\sqrt 2 + \sqrt 3 + \sqrt 5$ is $\sqrt 2 + \sqrt 3 - \sqrt 5$ (as we are done for two variables now converted into three). The second conjugation of $\sqrt 2 + \sqrt 3 + \sqrt 5$ is $\sqrt 2 - \sqrt 3 + \sqrt 5$ similarly the final conjugation of $\sqrt 2 + \sqrt 3 + \sqrt 5$ is $\sqrt 2 - \sqrt 3 - \sqrt 5$(both are negative)
Hence, we have $\sqrt 2 + \sqrt 3 - \sqrt 5$, $\sqrt 2 - \sqrt 3 + \sqrt 5$, $\sqrt 2 - \sqrt 3 - \sqrt 5$ are the conjugations of the term $\sqrt 2 + \sqrt 3 + \sqrt 5$
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