rationalizing numerator and denominator?

Question

how do i rationalize the numerator or denominator if the problem is as such (could you also please explain conjugate also like on problem 2. how do you get the conjugate if its on the left side of the subtraction sign?):

1. 3 / ( sqroot 5 + sqroot 6 )
2. 5 / ( sqroot 14 - 2 )
3. (sqroot 6 - 3 ) / 4
4. (sqroot 5 + sqroot 6 ) / 3

thanks, much appreciation.

Answer 1

ahhh rationalizing the denominator, basically having no radicals in the denominator

i'll give you a hint: remember the difference of squares property....

e.g. (a + b)(a - b) = a^2 - b^2

for number 1, the best thing to do is multiply that expression by
(sqrt 5 - sqrt6), both top and bottom...

what will happen is that the denominator will have no radical terms..

e.g. (sqrt 5 - sqrt 6)(sqrt 5 + sqrt 6) = (sqrt 5) ^ 2 - (sqrt 5)(sqrt 6) +(sqrt 5)(sqrt 6) - (sqrt 6)^2
= 5 - 6
= - 1

so you're denominator will be -1...
the numerator will be 3(sqrt 5 - sqrt 6)

you rationalized expression will be

3(sqrt 5 - sqrt 6) / -1 or -3 (sqrt 5 - sqrt 6)

try the same thing with #2 remembering the difference of squares property

Answer 2

Remember that (a+b)*(a-b) = a^2 - b^2

1) Multiply numerator and denominator by [sqrt(5) - sqrt(6)]. The denominator becomes the 5 - 6, or -1, so the rationalized result is -3*[sqrt(5) - sqrt(6)]

2) same thing, except multiply by [sqrt(14) + 2] to get 14-4 in the denominator

Rationalize the numerators in 3 and 4 the same way: the numerators will come out -3 and -1 respectively

Answer 3

You rationalize as you indicated by multiplying top and bottom by the conjugate of the part you want to rationalize. It doesn't matter if the sqrt part is on the left or right. On #2, the conjugate is (sqrt 14 + 2).

Your numerator is 5(sqrt 14 + 2);
denominator is sqrt 14 squared - 4= 14-4 = 10

The 5 cancels with the 10 and you get (sqrt 14 + 2)/2

Answer 4

So, you multiply the denominator and the numerator by the square root of whatever the denominator is. So the first one would be... sqroot(5x-6)(5x+3)/5x+3 And...uh, ok I admit I have no idea where to go from there. You might have to foil the numerator, but I don't really know what else. I think that might be it.

Answer 5

Rationalizing Numerator

Answer 6

use the opposite sign of the term (numerator / denominator) you want to rationalize

e.g

3 / (sqr 5 + sqr 6)
= 3 / (sqr 5 + sqr 6) x [(sqr 5 - sqr 6) / (sqr 5 - sqr 6)], this won't change the value

= 3 (sqr 5 - sqr 6) / (5-6)
= 3 (sqr 5 - sqr 6) / -1
= 3 sqr 6 - 3 sqr 5