Question

The conjugate of $3 + \sqrt 5 $ is?
A. $3 - \sqrt 5 $
B. $3 + \sqrt 5 $
C. $\sqrt 3 + \sqrt 5 $
D. $\sqrt 3 - 5$

Answer 1

Hint: Recall the definition of the conjugate and find the conjugate of the given term by changing the sign of irrational term. In, the given number, 3 is the rational term and $\sqrt 5 $ is an irrational term. The conjugate will have the sign of the irrational term as negative for this question.

Complete step by step Answer:

We are given an irrational number, that is , $3 + \sqrt 5 $
Here, 3 is a rational term and $\sqrt 5 $ is an irrational term
And thus, the given term is a binomial term.
We have to find the conjugate of the given term. The product of the conjugate pair gives rational value.
If a number is $a + b$, then the conjugate of the number is $a - b$.
We calculate the conjugate of numbers as it helps in rationalizing irrational numbers.
When we multiply the number and its conjugate together, we can get the formula, $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$
Here, we have $a + b$ is equal to $3 + \sqrt 5 $
This implies that $a = 3$ and $b = \sqrt 5 $
We know that conjugate is of the form $a - b$, then the conjugate is $3 - \sqrt 5 $
Thus, the conjugate of $3 + \sqrt 5 $ is $3 - \sqrt 5 $.
Hence, option A is correct.

Note: Conjugate pair means that the numbers have the same magnitude but have a sign of one term different. Conjugate is possible only for binomial terms. Here, we have $\sqrt 5 $ as an irrational number. Thus, the sign of $\sqrt 5 $ is different in the conjugate of the given expression. Also, the conjugate term is the rationalizing factor of the given term.