Question

What is the conjugate of binomial surd $5+\sqrt{3}$ ?
A. \[5-\sqrt{3}\]
B. \[3+\sqrt{5}\]
C. \[\sqrt{5}+\sqrt{3}\]
D. \[\sqrt{5}-\sqrt{3}\]

Answer 1

Hint: We know that surds are the square roots of numbers which cannot be simplified into a whole or rational number and cannot be represented in a fraction. Binomial Surd is an algebraic sum of two surds or a surd and a rational number. To find the conjugate of binomial surd $5+\sqrt{3}$ , we will be changing the ‘+’ sign, that spits the terms, into ‘-’ sign.

Complete step-by-step answer:
We need to find the conjugate of binomial surd $5+\sqrt{3}$ . Let us first see what a surd is.
Surds are the square roots of numbers which cannot be simplified into a whole or rational number and cannot be represented in a fraction. It is a root of the whole number that has an irrational value. For example, $\sqrt{3}=1.732050808$ . The value of $\sqrt{3}$ will be accurate when it is not written in decimal.
Now, let us see the definition of binomial surge.
Binomial Surd is an algebraic sum of two surds or a surd and a rational number. For example, $\sqrt{3}+\sqrt{8}\text{ and }4+\sqrt{5}$ .
From the above definition, we know that the given surd, that is, $5+\sqrt{3}$ is binomial.
Now, to find the conjugate of binomial surds, we will be changing the sign in between the terms, that is, ‘+’ becomes ‘-’ and ‘-’ becomes ‘+’.
Hence, the conjugate of $5+\sqrt{3}$ is $5-\sqrt{3}$ .

So, the correct answer is “Option A”.

Note: To check whether the answer $5-\sqrt{3}$ is the conjugate of $5+\sqrt{3}$ , we can do the following operations.
Let us multiply $5+\sqrt{3}$ with $5-\sqrt{3}$ .
$\Rightarrow \left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)$
We know that $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$ . Hence, the above equation becomes
$\left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)=\left( {{5}^{2}}-{{\left( \sqrt{3} \right)}^{2}} \right)$
Let us solve this. We will get
$\begin{align}
  & \left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)=\left( 25-3 \right) \\
 & \Rightarrow \left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)=22 \\
\end{align}$
We know that 22 is a rational number. So, when we multiply a binomial surd with its conjugate, we must get a rational number.