Question

$\sqrt{12}$ and _____ are like surds.
This question has multiple correct options.
A) $\sqrt{24}$
B)$\sqrt{36}$
C)$\sqrt{48}$
D) $\sqrt{60}$

Answer 1

Hint: To obtain the answer of the statement we will use the definition of surds. Firstly we will write what surds it is, then according to our definition we will check which among the options given to satisfy it and get our desired answer.

Complete step by step answer:
It is given to us that $\sqrt{12}$ is a surd.
So as we know that surds are the value in square root that cannot be simplified further to get a whole number or integers.
We will check each of the options and see which among them we can’t find a perfect square root of.
A) $\sqrt{24}$
For finding the square root of 24 we will find its factors and get the one in power of 2 outside the sign as follows:
$\begin{align}
  & \sqrt{24}=\sqrt{2\times 2\times 2\times 3} \\
 & \Rightarrow \sqrt{24}=\sqrt{{{2}^{2}}\times 2\times 3} \\
 & \Rightarrow \sqrt{24}=2\sqrt{2\times 3} \\
 & \therefore \sqrt{24}=2\sqrt{6} \\
\end{align}$
So as we can see that the square root of 24 is not a whole number or an integer.
Therefore $\sqrt{24}$ is a surd.
B) $\sqrt{36}$
$\begin{align}
  & \sqrt{36}=\sqrt{2\times 2\times 3\times 3} \\
 & \Rightarrow \sqrt{36}=\sqrt{{{2}^{2}}\times {{3}^{2}}} \\
 & \Rightarrow \sqrt{36}=2\times 3 \\
 & \therefore \sqrt{36}=6 \\
\end{align}$
So as we can see that the square root of 36 is a whole number.
Therefore $\sqrt{36}$ is not a surd
C)$\sqrt{48}$
$\begin{align}
  & \sqrt{48}=\sqrt{2\times 2\times 2\times 2\times 3} \\
 & \Rightarrow \sqrt{48}=\sqrt{{{2}^{2}}\times {{2}^{2}}\times 3} \\
 & \Rightarrow \sqrt{48}=2\times 2\sqrt{3} \\
 & \therefore \sqrt{48}=4\sqrt{3} \\
\end{align}$
So as we can see that the square root of 48 is not a whole number or an integer.
Therefore $\sqrt{48}$ is a surd.
D) $\sqrt{60}$
$\begin{align}
  & \sqrt{60}=\sqrt{2\times 2\times 3\times 5} \\
 & \Rightarrow \sqrt{60}=\sqrt{{{2}^{2}}\times 3\times 5} \\
 & \Rightarrow \sqrt{60}=2\sqrt{3\times 5} \\
 & \therefore \sqrt{60}=2\sqrt{15} \\
\end{align}$
So as we can see that the square root of 60 is not a whole number or an integer.
Therefore $\sqrt{60}$ is a surd.
Hence option (A), (C), (D) is correct.

Note: Surds are the square root of any number that cannot be simplified into a whole or rational number. Surds can’t be accurately represented in a fraction. A surd is a root of the whole number that has an irrational value. There are many types of surds namely Simple surds, Pure surds, Similar Surds, Mixed surds, Compound Surds, Binomial Surds etc.