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**Hint:**In this question, we will use factorization, and expansion of algebraic identities. For this problem, we will use the algebraic identity ${(a + b)^2} = {a^2} + 2ab + {b^2}$ .

**Complete step by step solution:**

Now, in this question, we have to find the square root of $12\sqrt 5 + 2\sqrt {55} $.

So it will become: $\sqrt {12\sqrt 5 + 2\sqrt {55} } $, which on simplification will become:

$

\sqrt {12\sqrt 5 + 2\sqrt {55} } \\

= \sqrt {12\sqrt 5 + 2\sqrt {5 \times 11} } \\

= \sqrt {12\sqrt 5 + 2\sqrt 5 .\sqrt {11} } \\

$

Now, to solve $\sqrt {12\sqrt 5 + 2\sqrt {55} } $, we will take $\sqrt 5 $ common within the under root and get:

$

= \sqrt {12\sqrt 5 + 2\sqrt 5 .\sqrt {11} } \\

= \sqrt {\sqrt 5 (12 + 2\sqrt {11} )} \\

$

Now we will change the term $(12 + 2\sqrt {11} )$ inside the under root sign to express it in terms of ${(a + b)^2}$ .

Now we can write $(12 + 2\sqrt {11} )$ as:

$(1 + 11 + 2\sqrt {11} )$ which can we reframed as:

$({1^2} + {(\sqrt {11} )^2} + 2\sqrt {11} )$, comparing it with the RHS of the expansion of the algebraic identity ${(a + b)^2}$ which is given as:

${a^2} + 2ab + {b^2}$

We will get $({1^2} + {(\sqrt {11} )^2} + 2\sqrt {11} )$= ${a^2} + 2ab + {b^2}$

So that,

$

{a^2} = {1^2}, \\

2ab = 2.1.\sqrt {11} \\

{b^2} = {(\sqrt {11} )^2} \\

$

Such that we get :

$

a = 1, \\

2ab = 2.1.\sqrt {11} \\

b = \sqrt {11} \\

$

Now, since

${a^2} + 2ab + {b^2} = {(a + b)^2}$

Then putting the values obtained above:

$({1^2} + {(\sqrt {11} )^2} + 2\sqrt {11} ) = {(1 + \sqrt {11} )^2}$

Therefore $\sqrt {12\sqrt 5 + 2\sqrt {55} } $ will now become:

$

= \sqrt {12\sqrt 5 + 2\sqrt 5 .\sqrt {11} } \\

= \sqrt {\sqrt 5 (12 + 2\sqrt {11} )} \\

= \sqrt {\sqrt 5 ({1^2} + {{(\sqrt {11} )}^2} + 2\sqrt {11} )} \\

= \sqrt {\sqrt 5 {{(1 + \sqrt {11} )}^2}} \\

= \sqrt {\sqrt 5 } (1 + \sqrt {11} ) \\

= \sqrt[4]{5}(1 + \sqrt {11} ) \\

$

So, finally we can say that :

Square root of $12\sqrt 5 + 2\sqrt {55} $

$ = \sqrt[4]{5}(\sqrt {11} + 1)$

**Hence, the correct answer is option A.**

**Note:**We cannot afford to forget the square root operation throughout the solution of this problem. For such problems, which require us to find the square root of another square root, we need to identify the algebraic expansion accurately so that we can get the correct corresponding algebraic identity to simplify and evaluate the square root.

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