Question

Find the square of $\sqrt {20 + \sqrt 3 + \sqrt {28 - 10\sqrt 3 } } $

Answer 1

Hint: To find the square of the above, we need to first see whether the given format can be in the form of the square number with which we can simplify that to get the required solution, the reason we are doing so is that they easily simplified

Complete step-by-step answer:
The main concept that we are going to follow is to form a perfect square in the root so that it can easily be out of the root because it will have 2 in the power.
So, let us consider the given $\sqrt {20 + \sqrt 3 + \sqrt {28 - 10\sqrt 3 } } $be $\sqrt {20 + \sqrt 3 + a} $
So, that we can say that $a = \sqrt {28 - 10\sqrt 3 } $
Now, let us first simplify the value of a, so that we can see the factors in a
$a = \sqrt {28 - 10\sqrt 3 } $
Let us see how it can form a perfect square so, that it can easily comes out of the root part
In the given case, we break 28 as shown below so that it can form a perfect square
$28 = 25 + 3 = {(5)^2} + {(\sqrt 3 )^2}$
Now the other part of a , can be break in the multiplication so that together with the first part it can make a perfect square
 $10\sqrt 3 = (2)(5)(\sqrt 3 )$
Now, substituting the required value of $28$ and $10\sqrt 3 $in a, we get
$a = \sqrt {28 - 10\sqrt 3 } $
$a = \sqrt {{{(5)}^2} + {{(\sqrt 3 )}^2} - 2(5)(\sqrt 3 )} $
Using the formula ${(a - b)^2} = {a^2} + {b^2} - 2(a)(b)$, Hence we obtain the perfect square now it can easily be simplified
So, the value of a
$a = \sqrt {{{(5)}^2} + {{(\sqrt 3 )}^2} - 2(5)(\sqrt 3 )} = \sqrt {{{(5 - \sqrt 3 )}^2}} = 5 - \sqrt 3 $
After getting the value of a, we can substitute the value in $\sqrt {20 + \sqrt 3 + a} $
$\sqrt {20 + \sqrt 3 + a} = \sqrt {20 + \sqrt 3 + 5 - \sqrt 3 } $
After simplification, we get
$\sqrt {20 + \sqrt 3 + \sqrt {28 - 10\sqrt 3 } } = \sqrt {20 + 5} = \sqrt {25} = 5$
The squaring of the above provides us with a result equal to 25.
${(\sqrt {20 + \sqrt 3 + \sqrt {28 - 10\sqrt 3 } } )^2} = {(\sqrt {20 + 5} )^2} = {(\sqrt {25} )^2} = {(5)^2} = 25$
So, 25 is the required result.

Note: In such type of equation the trick is to form a perfect square so that the given question can be easily simplified because in such case it comes out of a root because it have a square in the power, similarly in the above question the perfect square is forming which somehow helped to get the result easily.