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

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Last updated date: 14th Jul 2024
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Answer
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Hint: First we have to define what the terms are we need to solve in the problem. There are a total of three quadratics that can be written as the addition or from the difference too. A surd is an expression that includes a root whether square or cube or any roots; they are used to represent irrational numbers.

Complete step-by-step solution:
Surd means which are the numbers that will not be expressed as fractions or recurring decimals.
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 $.
Hence combining all the conjugates into three variables we get
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 $

Note: Since for the two variables conjugation we only need one result, for three variables we need two or three results, hence for n-variables conjugation we need at most n-results as the conjugation (as per the conjugation result).