Question

Find the square root of \[5 - 2\sqrt 6 \].
a) \[\sqrt {13} - \sqrt 2 \]
b) \[\sqrt 3 - \sqrt 2 \]
c) \[\sqrt 5 - \sqrt 3 \]
d) \[\sqrt 5 - \sqrt 2 \]

Answer 1

Hint: We will first decompose both the terms by writing first term as a sum of two number and then try to write it in the form of \[{a^2} + {b^2} - 2ab\] which is equal to \[{\left( {a - b} \right)^2}\]. We will then square root both sides to get the square root of the required expression and when we Square root the square term, we get \[\sqrt {{x^2}} = \pm x\]. We will then check for the options and choose from the given options.

Complete step by step solution:
We need to find the square root of \[5 - 2\sqrt 6 \] i.e. \[\sqrt {5 - 2\sqrt 6 } \].
First of all, we will try to write \[5 - 2\sqrt 6 \] on the form of \[{a^2} + {b^2} - 2ab\]
We will decompose \[5\] as \[5 = 3 + 2 - - - - - - (1)\]
We know, \[\sqrt a \times \sqrt b = \sqrt {ab} \]
And \[2\sqrt 6 \] as \[2\sqrt 6 = 2 \times \sqrt 3 \times \sqrt 2 - - - - - - (2)\]
Hence, using (1) and (2)
\[5 - 2\sqrt 6 = 3 + 2 - \left( {2 \times \sqrt 3 \times \sqrt 2 } \right)\]
We know that \[x = {\left( {\sqrt x } \right)^2}\]. So, we can write the above equations as :
\[5 - 2\sqrt 6 = 3 + 2 - \left( {2 \times \sqrt 3 \times \sqrt 2 } \right)\]
\[ = {\left( {\sqrt 3 } \right)^2} + {\left( {\sqrt 2 } \right)^2} - 2 \times \sqrt 3 \times \sqrt 2 \], which is of the form \[{\left( a \right)^2} + {\left( b \right)^2} - 2 \times a \times b\]
We know, \[{\left( a \right)^2} + {\left( b \right)^2} - 2 \times a \times b = {\left( {a - b} \right)^2}\]. So,
\[5 - 2\sqrt 6 = {\left( {\sqrt 3 } \right)^2} + {\left( {\sqrt 2 } \right)^2} - 2 \times \sqrt 3 \times \sqrt 2 \]
\[ = {\left( {\sqrt 3 - \sqrt 2 } \right)^2}\]
Hence, we have
\[5 - 2\sqrt 6 = {\left( {\sqrt 3 - \sqrt 2 } \right)^2}\]
Now, square rooting both the sides, we get
\[\sqrt {5 - 2\sqrt 6 } = \sqrt {{{\left( {\sqrt 3 - \sqrt 2 } \right)}^2}} \]
Using \[\sqrt {{a^2}} = \pm a\], we have
\[\sqrt {5 - 2\sqrt 6 } = \sqrt {{{\left( {\sqrt 3 - \sqrt 2 } \right)}^2}} = \pm \left( {\sqrt 3 - \sqrt 2 } \right)\]
So, we got
\[\sqrt {5 - 2\sqrt 6 } = \pm \left( {\sqrt 3 - \sqrt 2 } \right)\]
Taking two cases,
\[\sqrt {5 - 2\sqrt 6 } = \left( {\sqrt 3 - \sqrt 2 } \right)\] and \[\sqrt {5 - 2\sqrt 6 } = - \left( {\sqrt 3 - \sqrt 2 } \right)\]
\[ \Rightarrow \sqrt {5 - 2\sqrt 6 } = \left( {\sqrt 3 - \sqrt 2 } \right)\] and \[\sqrt {5 - 2\sqrt 6 } = \left( {\sqrt 2 - \sqrt 3 } \right)\]
Now, going to the options. We see that in the options, Option b) is \[\sqrt 3 - \sqrt 2 \] but none of the options is equal to \[\sqrt 2 - \sqrt 3 \].
So, the correct answer is “Option B”.

Note: We should take care of the sign in between as according to the sign, we will select whether we should write the given expression as \[{a^2} + {b^2} + 2ab\] or \[{a^2} + {b^2} - 2ab\]. When we do the square rooting, we have to keep in mind that both plus and minus signs should be there. We could have decomposed \[5\] as sum of any two numbers but as we know prime factors of \[6\] are \[2\] and \[3\] and \[\sqrt 6 = \sqrt 3 \times \sqrt 2 \], so we have to decompose \[5\] in such a way that we can write the expression in the form of \[{a^2} + {b^2} - 2ab\].