Question

Rationalise \[\dfrac{{\sqrt 3 {\text{ }} - {\text{ }}1}}{{\sqrt 3 {\text{ }} + {\text{ }}1}}\].

Answer 1

Hint: We try to find out the basic structure then we come to know that there are two structures but with different signs like \[a + b\] and \[a - b\] .In this case, we multiply the term with another term then it give rise to a new form that is \[\left( {a + b} \right)\left( {a - b} \right)\] or \[\left( {a - b} \right)\left( {a + b} \right)\] which is a identity that gives \[{a^2} - {b^2}\] as a result. These square terms eliminate the terms with root in power which makes the equation in simplest form which is the required answer. In other words, to rationalize the given equation, find the conjugate of the denominator. Then multiply the numerator and denominator of the given equation by the conjugate of the denominator to simplify the radicals in the denominator. Then solve the fraction further if needed to get the answer.

Complete step-by-step answer:
The given equation is \[\dfrac{{\sqrt 3 {\text{ }} - {\text{ }}1}}{{\sqrt 3 {\text{ }} + {\text{ }}1}}\] . To rationalize the given equation, on multiply and divide the denominator by \[\sqrt 3 {\text{ }} - {\text{ }}1\] we get
\[\dfrac{{\sqrt 3 {\text{ }} - {\text{ }}1}}{{\sqrt 3 {\text{ }} + {\text{ }}1}}\] \[ \times \] \[\dfrac{{\sqrt 3 {\text{ }} - {\text{ 1}}}}{{\sqrt 3 {\text{ }} - {\text{ }}1}}\]
On multiplying the terms in numerator and denominator we get
\[ = {\text{ }}\dfrac{{{{\left( {\sqrt 3 {\text{ }} - {\text{ }}1} \right)}^2}}}{{{{\left( {\sqrt 3 } \right)}^2}{\text{ }} - {\text{ }}{{\left( 1 \right)}^2}}}\]
As we know that \[{\left( {a - b} \right)^2} = {\text{ }}{a^2} + {\text{ }}{b^2} - {\text{ }}2ab\] . Therefore, on applying this identity in the numerator we get
\[ = {\text{ }}\dfrac{{3{\text{ }} + {\text{ }}1{\text{ }} - {\text{ }}2\left( {\sqrt 3 } \right)}}{{3{\text{ }} - {\text{ }}1}}\]
 \[ = {\text{ }}\dfrac{{4 - 2\sqrt 3 }}{2}\]
 \[ = {\text{ }}\dfrac{4}{2} - \dfrac{{2\sqrt 3 }}{2}\]
 \[ = {\text{ }}2 - \sqrt 3 \]
Hence, on rationalizing the given equation we get \[2 - \sqrt 3 \] .
So, the correct answer is “ \[2 - \sqrt 3 \]”.

Note: \[\sqrt 3 {\text{ }} - {\text{ }}1\] is the conjugate of \[\sqrt 3 + 1\] . To find the conjugate of the term we just have to change the sign that is between the two terms of the denominator but we have to keep the same order of the terms. For example, \[a + b\] and \[a - b\] are the conjugates of each other. Here in this question there are two terms that’s why we are taking the conjugate of the denominator. We rationalize the denominator to get rid of any radicals present in the denominator. It is important to simplify all the radicals.