Question

If \[x = \sqrt {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \] then calculate the value of \[{x^2}{(x - 4)^2} \] .
A. 7
B. 4
C. 2
D. 1

Answer 1

Hint: We will rationalize the given term. The conjugate surd of \[2 - \sqrt 3 \] is \[2 + \sqrt 3 \]. After that we will put the value of in the given expression.

Formula used: If we have a surd \[a + \sqrt b \] then its conjugate surd is \[a - \sqrt b \] and also vice-versa. I we have a fraction \[\dfrac{{c + \sqrt d }}{{a - \sqrt b }}\] then we always do rationalize and to rationalize this type of fraction we use the technique \[\dfrac{{(c + \sqrt d )(a + \sqrt {b)} }}{{(a - \sqrt b )(a + \sqrt {b)} }} = \dfrac{{(c + \sqrt d )(a + \sqrt {b)} }}{{{a^2} - b}}\] multiplying denominator and numerator by the conjugate surd of the denominator part to denominator as rational free term.
\[(a + b)(a - b) = {a^2} - {b^2}\].

Complete step-by-step solution:
We will rationalize the given term. The conjugate surd of \[2 - \sqrt 3 \] is \[2 + \sqrt 3 \].
We have
\[x = \sqrt {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \]
or, \[x = \sqrt {\dfrac{{(2 + \sqrt 3 )(2 + \sqrt 3 )}}{{(2 - \sqrt 3 )(2 + \sqrt 3 )}}} \] [Rationalizing by multiplication of \[2 + \sqrt 3 \] with numerator and denominator]
or, \[x = \sqrt {\dfrac{{{{(2 + \sqrt 3 )}^2}}}{{{2^2} - {{(\sqrt 3 )}^2}}}} \] [Using \[(a + b)(a - b) = {a^2} - {b^2}\] ]
or, \[x = \sqrt {\dfrac{{{{(2 + \sqrt 3 )}^2}}}{{4 - 3}}} \]
or, \[x = \sqrt {{{(2 + \sqrt 3 )}^2}} \]
or, \[x = 2 + \sqrt 3 \] [Taking positive square root]
Now, subtracting 4 from x we have
\[x - 4 = 2 + \sqrt 3 - 4 = - 2 + \sqrt 3 \]
So, \[{x^2}{(x - 4)^2} = {(2 + \sqrt 3 )^2}{( - 2 + \sqrt 3 )^2}\]
\[ = {(2 + \sqrt 3 )^2}{(2 - \sqrt 3 )^2}\]
\[ = {({2^2} - {(\sqrt 3 )^2})^2}\] [Using\[(a + b)(a - b) = {a^2} - {b^2}\]]
\[ = {(4 - 3)^2}\]
\[ = 1\]

Hence the option D. is correct.

Note: If we directly place the value of x in the desired term then the process will be very lengthy to find the answer. If we take a square of x directly then we have a very tough situation to handle this problem. Most of the students do this and have to face this difficult situation. Then they become puzzled to solve. So, first notice the question that you have to find, then following that you have to precede the solution to get an answer quickly. In this type of problem, first, we have to get rational free denominator terms using some techniques. The numerator term may be any term. But there should not be any surd term in the denominator.