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**Hint:**First we have to define what the terms we need to solve the problem are.

First write the square root of $1.96$ the number into like $\dfrac{{196}}{{100}}$ $ \times 100$(so that the hundred will be cancelled each other) so the values of the given question will not be changed.

Also, since we will not be going to use the calculator for this solving method

**Complete step by step answer:**

First of all, take the square root of the given number which is $\sqrt {1.96} $(square root means the common terms in the natural numbers can be taken out) and the root of the terms is the multiplied times the given number twice.

Change the square root as the pure natural numbers in the numerator; \[\sqrt {\dfrac{{1.96}}{{100}} \times 100} \]

Now multiply the hundred into $1.96$times we get \[\sqrt {\dfrac{{196}}{{100}}} \]now taking the common terms out first in the denominator we get $\sqrt {100} = 10$and substitute we get \[\sqrt {\dfrac{{196}}{{100}}} = \dfrac{{\sqrt {196} }}{{10}}\]similarly we can rewrite the numerator as the pure gettable out form as $196 = 4 \times 49$

Since seven squares is the forty-nine and also two squares are the four and inverse form of the square is the square root; hence we get \[\dfrac{{\sqrt {4 \times 49} }}{{10}}\]now taking the common terms out as like as above derivations

Thus, we get \[\dfrac{{\sqrt {4 \times 49} }}{{10}} = \dfrac{{2 \times 7}}{{10}}\]hence now multiply the numerator terms and using division law we will divide further; $\dfrac{{14}}{{10}} = 1.4$is the required square root of $1.96$without a calculator

Also if we square ${(1.4)^2}$we will get the exact$1.96$, so only we say that square and square roots are inverse forms.

**Note:**Since the given question ask us to find the square root without using the purpose of the calculator so only, we change that into the natural form and then taking the common terms out and cancelled we get the answer, if the calculator can be usable the answer is in one step only, which is $\sqrt {1.96} $= $1.4$

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