Question

Find the value of \[a\] and \[b\] in each of the following:
 \[\dfrac{{\sqrt 2 + \sqrt 3 }}{{3\sqrt 2 - 2\sqrt 3 }} = a - b\sqrt 6 \]

Answer 1

Hint: In order to solve this question, we have to rationalize the number because to form an integer at the denominator. If \[a + b\] is given in the numerator then on rationalization we have to multiply and divide by \[a - b\] . and after rationalization, we have to compare the right-hand side and left-hand side to get the values of \[a\] and \[b\] .

Complete step-by-step answer:
Given,
A equation
 \[\dfrac{{\sqrt 2 + \sqrt 3 }}{{3\sqrt 2 - 2\sqrt 3 }} = a - b\sqrt 6 \]
To find,
The value of \[a\] and \[b\] in the given equation.
To solve this question we have to rationalize this number to get an integer in the denominator.
First we take left and side
 \[ = \dfrac{{\sqrt 2 + \sqrt 3 }}{{3\sqrt 2 - 2\sqrt 3 }}\]
After multiplying and divide by \[3\sqrt 2 + 2\sqrt 3 \] we get,
 \[ = \dfrac{{\sqrt 2 + \sqrt 3 }}{{3\sqrt 2 - 2\sqrt 3 }} \times \dfrac{{3\sqrt 2 + 2\sqrt 3 }}{{3\sqrt 2 + 2\sqrt 3 }}\]
On arranging this equation
 \[ = \dfrac{{\left( {\sqrt 2 + \sqrt 3 } \right) \times \left( {3\sqrt 2 + 2\sqrt 3 } \right)}}{{\left( {3\sqrt 2 - 2\sqrt 3 } \right) \times \left( {3\sqrt 2 + 2\sqrt 3 } \right)}}\]
After multiplying in nominator
 \[ = \dfrac{{\left( {3\sqrt 2 \times \sqrt 2 + 2\sqrt 3 \times \sqrt 2 + 3\sqrt 2 \times \sqrt 3 + 2\sqrt 3 \times \sqrt 3 } \right)}}{{\left( {3\sqrt 2 - 2\sqrt 3 } \right) \times \left( {3\sqrt 2 + 2\sqrt 3 } \right)}}\]
On further solving
 \[ = \dfrac{{\left( {6 + 2\sqrt 6 + 3\sqrt 6 + 6} \right)}}{{\left( {3\sqrt 2 - 2\sqrt 3 } \right) \times \left( {3\sqrt 2 + 2\sqrt 3 } \right)}}\]
On applying the identity of \[\left( {a + b} \right) \times \left( {a - b} \right) = {a^2} - {b^2}\] in denominator
 \[ = \dfrac{{\left( {12 + 5\sqrt 6 } \right)}}{{\left( {{{\left( {3\sqrt 2 } \right)}^2} - {{\left( {2\sqrt 3 } \right)}^2}} \right)}}\]
On further solving
 \[ = \dfrac{{\left( {12 + 5\sqrt 6 } \right)}}{{\left( {18 - 12} \right)}}\]
 \[ = \dfrac{{\left( {12 + 5\sqrt 6 } \right)}}{{\left( {6} \right)}}\]
On separating
 \[ = 2 + \dfrac{{5\sqrt 6 }}{{6}}\]
On solving left hand side the value of left hand side is
 \[LHS = 2 + \dfrac{{5\sqrt 6 }}{{6}}\]
on equating with right hand side
 \[2 + \dfrac{{5\sqrt 6 }}{{6}} = a - b\sqrt 6 \]
On comparing both side we find the value of \[a\] and \[b\]
On comparing term \[\sqrt 6 \] and another term separately we get the values of \[a\] and \[b\] .
 \[a = 2\] and
 \[b = -\dfrac{5}{{6}}\]

Note: To solve this type of question you must have knowledge of rationalizing the number and after solving that side try to compare that side with the other side and try to find the value of asked values. You may commit mistakes in rationalizing and comparing those values and find the values that are asked in the question.