Question

2+√(−3)/4+√(−3)
Express the following in the form of a+ib, a, beR i=√(-1)

Answer 1

Answer :

$\large{\star\:\:\boxed{{\dfrac{11}{19}\:+\:\dfrac{2\sqrt{3}}{19}i}}\:\:\star}$

Explanation :

Given :–

• $\dfrac{2\:+\sqrt{-3} }{4\:+\:\sqrt{-3} }$ is a Complex Number .

• $i\:=\:\sqrt{-1}$

To Express :–

We have to express $\dfrac{2\:+\sqrt{-3} }{4\:+\:\sqrt{-3} }$ in the form of a + ib .

Formula Used :–

• $\boxed{\bf{\star\:\:(a\:+\:b)(a\:-\:b)\:=\:a^2\:-\:b^2\:\:\star}}$

Solution :–

We have ,

$\implies\dfrac{2\:+\:\sqrt{-3} }{4\:+\:\sqrt{-3} }$

$\implies\dfrac{2\:+\:\sqrt{(-1)(3)} }{4\:+\:\sqrt{(-1)(3)} }$

$\implies\dfrac{2\:+\:(\sqrt{-1})\sqrt{3} }{4\:+\:(\sqrt{-1})\sqrt{3} }$

Now , we know i = √(-1) .

Putting √(-1) as i :

$\implies\dfrac{2\:+\:\sqrt{3}i }{4\:+\:\sqrt{3}i }$

Now we will Rationalize the Denominator :

$\implies\dfrac{(2\:+\:\sqrt{3}i)(4\:-\:\sqrt{3}i) }{(4\:+\:\sqrt{3}i)(4\:-\:\sqrt{3}i) }$

$\implies\dfrac{8\:-\:2\sqrt{3}i\:+\:4\sqrt{3}i\:-\:(\sqrt{3}i)^2 }{(4)^2\:-\:(\sqrt{3}i)^2 }$

$\implies\dfrac{8\:+\:2\sqrt{3}i\:-\:3i^2 }{16\:-\:3i^2 }\:\:-----\bf{(1)}$

Now we know that ,  

i = √(-1)

Squaring Both Sides

i² = (-1)

Putting this value of i² in Equation(1) :-

$\implies\dfrac{8\:+\:2\sqrt{3}i\:-\:3(-1) }{16\:-\:3(-1) }$

$\implies\dfrac{8\:+\:2\sqrt{3}i\:+\:3 }{16\:+\:3 }$

$\implies\dfrac{11\:+\:2\sqrt{3}i}{19}$

Now separating both Terms :-

$\implies\dfrac{11}{19}\:+\:\dfrac{2\sqrt{3}i}{19}$

So we can write it as :

$\implies(\dfrac{11}{19})\:+\:(\dfrac{2\sqrt{3}}{19})i$

where, $\bf{a\:=\:\dfrac{11}{19}\:and\:b\:=\:\dfrac{2\sqrt{3}}{19}}$ .

Answer 2

$\huge{\mathfrak{\underline{\red{Answer!}}}}$

$\frac{2 + \sqrt{3} \: i }{4 + \sqrt{3} \: i } = \frac{(2 + \sqrt{3}i)(4 - \sqrt{3}i) }{(4 + \sqrt{3}i)(4 - \sqrt{3}i) } =$

$\frac{8 + 3 + 2 \sqrt{3}i }{19} = \frac{11}{19} + \frac{2 \sqrt{3} }{19} i$