Question

If a and b is rational number and \[\dfrac{3 + \sqrt{7}}{\left( 3 - \sqrt{7} \right)} = \ a + b\sqrt{}7\] .Find the value of \[a\] and \[b\].

Answer 1

Hint: In this question, we have been given that \[\dfrac{3 + \sqrt{7}}{\left( 3 - \sqrt{7} \right)} = \ a + b\sqrt{7}\] . We need to find the value of \[a\] and \[b\]. In order to find the value of \[a\] and \[b\], we need to use algebraic identities. First, we need to multiply with the conjugate term of the expression. Then by proceeding we can get the value of \[a\] and \[b.\]
Formula used :
\[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]
\[\left( a – b \right)^{2} = a^{2} + b^{2} – 2ab\]
\[a^{2} – b^{2} = \left( a + b \right)\left( a – b \right)\]

Complete step-by-step solution:
Given,
\[\dfrac{3 + \sqrt{7}}{\left( 3 - \sqrt{7} \right)} = \ a + b\sqrt{7}\]
In order to rationalize this , we need to multiply the left side term with its conjugate .
The conjugate term of \[3 - \sqrt{7}\ \]is \[\left( 3 + \sqrt{7} \right)\]
\[\dfrac{3 + \sqrt{7}}{\left( 3 - \sqrt{7} \right)} \times \dfrac{3 + \sqrt{7}}{3 + \sqrt{7}} = \ a + b\sqrt{7}\]
By multiplying,
We get,
\[\dfrac{\left( 3 + \sqrt{7} \right)^{2}}{\left( 3 - \sqrt{7} \right)\left( 3 + \sqrt{7} \right)} = \ a + b\sqrt{7}\]
We also know that,
\[a^{2} – b^{2} = \left( a + b \right)\left( a – b \right)\]
Thus it becomes,
\[\dfrac{\left( 3 + \sqrt{7} \right)^{2}}{3^{2} - \left( \sqrt{7} \right)^{2}} = a + b\sqrt{7}\]
We also know that, \[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]
Thus by expanding,
We get,
\[\dfrac{\left( 3 \right)^{2} + \left( \sqrt{7} \right)^{2} + 2\left( 3 \right)\left( \sqrt{7} \right)}{9 – 7} = a + b\sqrt{7}\]
By simplifying,
We get,
 \[\dfrac{9 + 7 + 6\sqrt{7}}{2} = a + b\sqrt{7}\]
\[\dfrac{16 + 6\sqrt{7}}{2} = a + b\sqrt{7}\]
Taking 2 as common ,
We get ,
 \[\dfrac{2\left( 8 + 3\sqrt{7} \right)}{2} = a + b\sqrt{7}\]
On further simplifying,
We get,
\[8 + 3\sqrt{7} = a + b\sqrt{7}\]
Now on comparing both sides,
We get,
\[a = 8\] and \[b = 3\]

The value of \[a\] is \[8\] and \[b\] is \[3\].

Note: The concept used to solve this problem is algebraic identities. Conjugate number is nothing but a number with the same magnitude but opposite sign. The simple example for this is the conjugate of \[\ (x+2) \] is \[\ (x-2) \] .The product of a number with its conjugate gives a real answer.