Write in simplified radical form by rationalizing the denominator: (sqrt(3) - 2) / (-5*sqrt(3)+8)?

Question

I'm pretty lost here. If someone can help please do.

Answer 1

(√3 - 2) / (-5) √3 + 8)

  It's been a litle while since I did this so I reviewed at a great site called purplemath.com (see reference below). This is what I got from it:

If you are rationalizing a denominator that consists of just the single radical (such as 2 / √3) you would multiply by √3/√3 and get 2√3/3 ... because 2 * √3 = 2√3 and √3* √3 = 3). But if you have more than one term (like you do) you have to multiply by the "conjugate" (it's that whole difference of squares thing where the middle term drops out and you are left with your 1st term squared minus your 2nd term squared) (Ex: 2 + √3  * 2 - √3 = (2)^2 - (√3)^2 = 4 - 3 = 1)

Getting back to your equation, (√3 - 2) / -5 √3 + 8)

Multiply by the conjugate of your denominator, ( - 5√3 - 8)(over itself) :

(√3 - 2) / (-5 √3 + 8)  * (- 5√3 - 8)/ (- 5√3 - 8) =

(√3 - 2) *  - 5√3 - 8 / (- 5√3 + 8) *(- 5√3 - 8) =

I use the FOIL method when doing two term multiplication (multiply 1st term x 1st term, outer x outer, inner x inner, then last term x last term and simplify)

(√3 *  - 5√3) = -5 (√3*√3) = -5(3) = -15

(√3 * - 8) = - 8√3

(- 2 *  - 5√3) = 10√3

(-2) * (-8) = 16

Now simplify the numerator: - 8√3 +10√3 -15 + 16 =

2√3 + 1

Multiply out the denominator (remember 1st term^2 - 2nd term ^2)

(- 5√3 + 8) *(- 5√3 - 8) = (- 5√3)^2 - (8)^2 =

(- 5√3)*(- 5√3) = 25 (√3*√3) = 25 * 3 = 75

(8)^2 = 64

Simplify the denominator: 75 - 64 = 11

Put your simplified numerator over your rationalized denominator --->

2√3 + 1/ 11 (Answer)

At least that's what I got ... I'm pretty sure it's right ... I hope a math genius will double-check me.

I hope I helped

Answer 2

(√3 - 2) / (-5√3 + 8)

Multiply by (-5√3 - 8)/(-5√3 - 8)

[ (√3 - 2)(-5√3 - 8) ] / [ (-5√3 + 8)(-5√3 - 8) ]

(-15 - 8√3 + 10√3 + 16) / (75 + 40√3 - 40√3 - 64)

(2√3 + 1) / 11