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**Hint:**To solve such questions that include surds and indices, a basic knowledge of the rules of surds is necessary. The question can be solved by evaluating the expression under the root and then by further simplifying the expression to get the final answer.

**Complete step by step answer:**

The given expression to simplify is $\dfrac{{\sqrt 4 }}{{36}}$ $...(i)$

Now we know that ${(2)^2} = 2 \times 2 = 4$ therefore,

$\sqrt 4 = {(4)^{1/2}}$

Which on further simplification gives us

${(4)^{1/2}} = {({2^2})^{1/2}}$

Applying the law of exponents that states ${({a^m})^n} = {a^{m \times n}}$ to the above expression,

${(4)^{1/2}} = {2^{2 \times \dfrac{1}{2}}}$

On simplifying the powers of $2$ we get

$\Rightarrow \sqrt 4 = 2$ $...(ii)$

Now substituting the value of the equation $(ii)$ in the equation $(i)$ we get,

$\Rightarrow \dfrac{{\sqrt 4 }}{{36}} = \dfrac{2}{{36}}$

Dividing and simplifying the above expression to get

$\Rightarrow \dfrac{1}{{18}}$

Hence, on simplifying $\dfrac{{\sqrt 4 }}{{36}}$ we get $\dfrac{1}{{18}}$

**Therefore, $\dfrac{{\sqrt 4 }}{{36}} = \dfrac{1}{{18}}$**

**Additional Information:**

A surd can be defined as an irrational number that can be expressed with roots, such as $\sqrt 2$ or $\sqrt[4]{{15}}$ . An index on the other hand can be defined as the power, or exponent, of a number. For example, ${2^3}$ has an index of $3$ . When we deal with exponents or powers of a number, a root often refers to a number that is repeatedly multiplied by itself a certain fixed number of times to get another number. A radical number can be written as shown below:

$\sqrt[n]{x}$ , where $n$ is the degree, and the root sign is known as the radical sign. The value $x$ is known as the radicand and the expression as a whole is known as the radical.

**Note:**While solving these types of questions it always proves extremely helpful if students remember the fundamental rules of surds and exponents. Some of the rules such as ${({a^m})^n} = {a^{m \times n}}$ and $\sqrt {a \times b} = \sqrt a \times \sqrt b$ are used a lot of times and help to simplify the question to a great extent.

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