Question

3+1/√3 add 1/3+√3 add 1/3-√3

Answer 1

Answer:

${ \sf{ \frac{3 + 1}{\sqrt{3} } + \frac{1}{3 + \sqrt{3} } + \frac{1}{3 - \sqrt{3} } }}$

${ \sf{ = \frac{4}{ \sqrt{3} } + \frac{(3 - \sqrt{3}) + (3 + \sqrt{3} ) }{(3 - \sqrt{3} )(3 + \sqrt{3}) } }}$

Using

(a + b)(a - b) = a² - b²

${ \sf{ = \frac{4}{ \sqrt{3} } + \frac{6}{ {3}^{2} - { \sqrt{3} }^{2} } }}$

${ \sf{ = \frac{4}{ \sqrt{3} } + \frac{6}{9 -0 } }}$

${ \sf{= \frac{4}{ \sqrt{3} } + \frac{ \cancel6}{ \cancel9} }}$

${ \sf{ = \frac{4}{ \sqrt{3} } + \frac{2}{3} }}$

$= \frac{12 + 2}{3 \sqrt{3} }$

$= \frac{14}{3 \sqrt{3} }$

So,

${ \bf{ \frac{3 + 1}{ \sqrt{3} } + \frac{1}{3 + \sqrt{3} } + \frac{1}{3 - \sqrt{3} } \to \frac{14}{3 \sqrt{3} } }}$

More information:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

(a + b)(a - b) = a² - b²

(a + b)² - (a - b)² = 4ab

(a + b)² + (a - b)² = 2(a² + b²)

Concepts used:

→ Surds

→ Rationalization