Answer
Verified
388.5k+ views
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.
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.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE