How does this simplify (trig)?

Question

[(sqrt3)+1] / 1-sqrt3

Can someone please show the steps how this becomes
(4+2sqrt3) / 1-3

I missed this problem on the test because I wasn't sure how to simplify

Also, can it be simplifed further than (4+2sqrt3) /1-3 ?? To me it seems like that should become 2+2sqrt3..am I correct or no?

Answer 1

( √3 + 1 ) / (1 - √3)

(√3 + 1)(1 + √3) / (1 - √3)(1 + √3)

(1 + 2√3 + 3) / (1 - 3)

(4 + 2√3) / (- 2)

- 2 - √3

Answer 2

(1 + sqrt3)/(1-sqrt3) is your expn
Multiply by conjugate (1 + sqrt3)/(1+sqrt3) ,

TIP: In other words, you multiply and divide your given expression by the SAME NUMBER. This is called "multiplying by the conjugate" Why do you do this? Just to simplify and get rid of the SQRT sign in the denominator.

Numerator = (1+sqrt3) ( 1 + sqrt3)
= 1*1 + sqrt3*aqrt3 + 2*sqrt3
= 1 + 3 + 2*sqrt3
= 4+2sqrt3

Denominator = (1 - sqrt3)*(1+sqrt3)
=( 1*1 - sqrt3*sqrt3)
= (1 -3)

Numerator/Denominator
= (4+2sqrt3)
------------
(1-3)

which is your answer that you have in your question.