Question

2. Express √5+√3i/ √5-√3i in the form of a + bi & state the values of a and b.​

Answer 1

ANSWER:

Given:

• (√5+√3i) / (√5-√3i)

To Do:

• Expressing the expression in the form a+bi.
• Find values of a and b.

Solution:

$\text{We are given that,}\\\\:\longrightarrow\dfrac{\sqrt5+\sqrt3i}{\sqrt5-\sqrt3i}\\\\\text{On rationalising,}\\\\:\implies\dfrac{\sqrt5+\sqrt3i}{\sqrt5-\sqrt3i}\times\dfrac{\sqrt5+\sqrt3i}{\sqrt5+\sqrt3i}\\\\:\implies\dfrac{(\sqrt5+\sqrt3i)\times(\sqrt5+\sqrt3i)}{(\sqrt5-\sqrt3i) \times(\sqrt5+\sqrt3i)}\\\\\text{We know that,}\\\\:\hookrightarrow(a+b)(a-b)=a^2-b^2\\\\\text{So,}\\\\:\implies\dfrac{(\sqrt5+\sqrt3i)^2}{(\sqrt5)^2-(\sqrt3i)^2}\\\\\text{We know that,}$

$:\hookrightarrow(a+b)^2=a^2+2ab+b^2\\\\\text{So,}\\\\:\implies\dfrac{(\sqrt5)^2+2(\sqrt5)(\sqrt3i)+(\sqrt3i)^2}{5-3i^2}\\\\:\implies\dfrac{5+2\sqrt{15}i+3i^2}{5-3i^2}\\\\\text{We know that,}\\\\:\hookrightarrow i^2=-1\\\\\text{So,}\\\\:\implies\dfrac{5+2\sqrt{15}i+3(-1)}{5-3(-1)}\\\\:\implies\dfrac{5+2\sqrt{15}i-3}{5+3}\\\\:\implies\dfrac{2+2\sqrt{15}i}{8}\\\\:\implies\dfrac{2\!\!\!/(1+\sqrt{15}i)}{8\!\!\!/_{\:4}}\\\\:\implies\dfrac{1+\sqrt{15}i}{4}\\\\\bf{:\implies\dfrac{1}{4}+\dfrac{\sqrt{15}}{4}i}$

$\text{Hence,}\\\\:\implies a+bi=\dfrac{1}{4}+\dfrac{\sqrt{15}}{4}i\\\\\text{Therefore, on comparing real and imaginary terms,}\\\\\bf{:\implies a=\dfrac{1}{4}\:\:\&\:\:b=\dfrac{\sqrt{15}}{4}}$

Formulae Used:

• (a + b)(a - b) = a² - b²
• (a + b)² = a² + 2ab + b²
• i² = -1