Question

Find the value of a and b if (√3+1)/(√3-1)=a+b√3

Answer 1

Question

Find the value of a and b , if

$\sf\dfrac{√3+1}{√3-1}=a+b√3$

Answer

$\colorbox{orange}{\sf \: a = 2 } \qquad \colorbox{cyan}{ \: \sf b = 1}$

Solution

Here, in order to find the values for a & b, we will take LHS first . In LHS, it's given a rational number in the form of surd. We will first rationalise the number to remove surd from its denominator, for rationalizing we have to multiply and divide the number with conjugates of denominator.

Let's Proceed with calculation !!

Taking L.H.S,

$:↦ \rm\dfrac{√3+1}{√3-1} \times \dfrac{√3+1}{√3 + 1}$

$:↦ \rm \dfrac{( \sqrt{3} + 1)^{2} }{( \sqrt{3}) ^{2} - (1)^{2} }$

$:↦ \sf\dfrac{( \sqrt{3})^{2} + 1 + 2 \sqrt{3} }{3 - 1}$

$:↦ \sf\dfrac{3 + 1 + 2 \sqrt{3} }{3 - 1}$

$:↦ \sf \dfrac{4 + 2 \sqrt{3} }{2}$

$:↦ \sf \dfrac{ \cancel2(2 + \sqrt{3})}{ \cancel2}$

$:↦ \sf2 + \sqrt{3}$

Now, just compare LHS with that of RHS

↦ 2 + √3 = a + b√3

↦ a = 2

↦ b = 1

$\rule{200pts}{2pt}$

★ Concept

Here , we are provided with a complex type of fraction in the form of p/q , which is a rational number but as it have a surd in its consequent (denominator) , the method 'Rationalization' will be used. For that , we have to multiply as well as divide the number with the conjugates of denominator ; conjugates means like if denominator is (a + b) , its conjugate will be (a - b) .

Thankyou