Question

Rationalize the denominator. simplify your answer. 5+sqrt3/2-sqrt3

Answer 1

`13 + 7√(3)`

Explanation:

Given

`(5 + √(3))/(2 - √(3))`

Required

Rationalize

To rationalize, we have to multiply the numerator and the denominator by
`{2 + √(3)}`

This gives

`(5 + √(3))/(2 - √(3)) * (2 + √(3))/(2 + √(3))`

`((5 + √(3))(2 + √(3)))/((2 - √(3))(2 + √(3)))`

Expand

`(5(2 + √(3))+√(3)(2 + √(3)))/(2(2 + √(3)) - √(3)(2 + √(3)))`

Open the brackets

`(10 + 5√(3)+2√(3) + (√(3))^2)/(4 + 2√(3) - 2√(3) - (√(3))^2)`

`(10 + 5√(3)+2√(3) + 3)/(4 + 2√(3) - 2√(3) - 3)`

Collect like terms for ease of arithmetic operations

`(10 + 3 + 5√(3)+2√(3))/(4 -3 + 2√(3) - 2√(3))`

`(13 + 7√(3))/(1 )`

`13 + 7√(3)`

Hence,

`(5 + √(3))/(2 - √(3))` is equivalent to
`13 + 7√(3)`, after rationalizing the denominator