Question

The simplified value of ${\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right)$ is

Answer 1

Hint: In this question use the direct algebraic identity of ${\left( {a + b} \right)^2}$which is${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. Perform operation to the terms that are like for example $a\sqrt b {\text{ and p}}\sqrt b $ can be simplified and not $a\sqrt b {\text{ and p}}\sqrt q $ as they are not like.

Complete step-by-step answer:
Given equation is
${\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right)$
Now expand the square according to property ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ we have,
$ \Rightarrow {\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right) = \left[ {{{\left( {\sqrt 3 } \right)}^2} + {1^2} + 2\sqrt 3 } \right] - 4 - 2\sqrt 3 $
Now simplify the above equation we have,
$ \Rightarrow {\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right) = \left[ {3 + 1 + 2\sqrt 3 } \right] - 4 - 2\sqrt 3 $
$ \Rightarrow {\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right) = 4 + 2\sqrt 3 - 4 - 2\sqrt 3 $
So as we see that R.H.S of the above equation is all canceled out.
$ \Rightarrow {\left( {\sqrt 3 + 1} \right)^2} - 2\left( {2 + \sqrt 3 } \right) = 0$
So this is the required simplified value of the given equation.

Note: These questions are solemnly based upon algebraic identity and pure simplification. It is always advised to remember basic algebraic identities like ${\left( {a + b} \right)^2}$ has been mentioned above, ${(a + b)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$. It helps to save a lot of time.