How do you perform the given division $\dfrac{\sqrt{144}}{\sqrt{36}}$?

Hint: We start solving the problem by equating the given division to a variable. We then factorize the numerator and make use of the fact that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$ to proceed through the problem. We then make the necessary calculations and make use of the fact that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to proceed further through the problem. We then make the necessary calculations to get the required answer for the given problem.

Complete step by step answer:
According to the problem, we are asked to find the result of the given division $\dfrac{\sqrt{144}}{\sqrt{36}}$.
Let us assume $d=\dfrac{\sqrt{144}}{\sqrt{36}}$.
$\Rightarrow d=\dfrac{{{\left( 144 \right)}^{\dfrac{1}{2}}}}{{{\left( 36 \right)}^{\dfrac{1}{2}}}}$ ---(1).
Let us first factorize the term given in the numerator to proceed through the problem.
We have $144=2\times 72=2\times 2\times 36={{2}^{2}}\times 36$ ---(2).
Let us substitute the result obtained from equation (2) in equation (1).
$\Rightarrow d=\dfrac{{{\left( {{2}^{2}}\times 36 \right)}^{\dfrac{1}{2}}}}{{{\left( 36 \right)}^{\dfrac{1}{2}}}}$ ---(3).
From laws of exponents, we know that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. Let us use this result in equation (3).
$\Rightarrow d=\dfrac{{{\left( {{2}^{2}} \right)}^{\dfrac{1}{2}}}\times {{\left( 36 \right)}^{\dfrac{1}{2}}}}{{{\left( 36 \right)}^{\dfrac{1}{2}}}}$.
$\Rightarrow d={{\left( {{2}^{2}} \right)}^{\dfrac{1}{2}}}$ ---(4).
From laws of exponents, we know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$. Let us use this result in equation (4).
$\Rightarrow d={{2}^{2\times \dfrac{1}{2}}}$.
$\Rightarrow d={{2}^{1}}$.
$\Rightarrow d=2$.
So, we have found the result of the given division $\dfrac{\sqrt{144}}{\sqrt{36}}$ as 2.

$\therefore $ The result of the given division $\dfrac{\sqrt{144}}{\sqrt{36}}$ is 2.

Note: We should perform each step carefully to avoid confusion and calculation mistakes. We can also solve this problem as shown below:
We have $d=\dfrac{\sqrt{144}}{\sqrt{36}}$ ---(5).
We know that $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$. Let us use this result in equation (5).
$\Rightarrow d=\sqrt{\dfrac{144}{36}}$.
$\Rightarrow d=\sqrt{4}$ ---(6).
We know that $\sqrt{4}=2$. Let us use this in equation (6).
$\Rightarrow d=2$.
$\Rightarrow \dfrac{\sqrt{144}}{\sqrt{36}}=2$.