Rationalize the denominator: 1/ (1+ sqt3 - sqt5)?

Question

I understand how to rationalize a binomial denominator but i need help rationalizing 1/ (1+ sqt3 - sqt 5)
ur earliest response is appreciated.. THANKS a bunch!

the answer is:
7+ 3sqt3 + sqt5 +2 sqt15 / 11
i just don't know how to arrive at the answer....

Answer 1

You need to multiply the denominator by something to get rid of the radicals.

Try multiplying by (1 - sqrt(3) + sqrt(5)), on the theory that it'll cancel out some of the radicals (like multiplying x+y by x-y to get x^2-y^2):

1/ (1+ sqrt(3) - sqrt(5)) =
(1 - sqrt(3) + sqrt(5)) / ((1 + sqrt(3) - sqrt(5)) * (1 - sqrt(3) + sqrt(5))) =

It gets a bit ugly here, it's like FOIL but using 3x3 terms gets 9 products in the denominator:

(1 - sqrt(3) + sqrt(5)) / (1*1 - 1*sqrt(3) + 1*sqrt(5) + sqrt(3)*1 - sqrt(3)*sqrt(3) + sqrt(3)*sqrt(5) - sqrt(5)*1 + sqrt(5)*sqrt(3) - sqrt(5)*sqrt(5)) =

At least the sqrt(3) and sqrt(5) terms all cancel:

(1 - sqrt(3) + sqrt(5)) / (1 - 3 - 5 + 2*sqrt(15)) =
(1 - sqrt(3) + sqrt(5)) / (2sqrt(15) - 7)

Now, the radicals are not all eliminated, but you can multiply the denominator that you now have (2sqrt(15) - 7), by (2sqrt(15) + 7), to get a difference of squares and eliminate the final radical in the denominator:

(1 - sqrt(3) + sqrt(5)) / (2sqrt(15) - 7)
(1 - sqrt(3) + sqrt(5)) * (2sqrt(15)+7) / ((2sqrt(15) - 7) * (2sqrt(15)+7))
(1 - sqrt(3) + sqrt(5)) * (2sqrt(15)+7) / (4*15 - 49)
(1 - sqrt(3) + sqrt(5)) * (2sqrt(15)+7) / 11

Now we can multiply out the numerator and combine like terms:

(1 - sqrt(3) + sqrt(5)) * (2sqrt(15)+7) / 11 =
(2sqrt(15) + 7 - 6sqrt(5) -7sqrt(3) + 10sqrt(3) + 7sqrt(5)) / 11 =
( 7 + 3sqrt(3) + sqrt(5) + 2sqrt(15) ) / 11

... which is the answer that you posted above. The subsequent poster did not assume the parentheses and only divided 2sqrt(15) by 11, it seems.

Answer 2

Umm, it looks like what you posted as the answer and the question are not the same.....

1/(1+sqrt(3)-sqrt(5)) =2.0162

7+3*sqrt(3)+sqrt(5) + 2*sqrt(15)/11 = 17.1364

But I would start by multiplying top and bottom by (1-sqrt(3)+sqrt(5)) this will result in a denominator of

(1+2sqrt(15)) which you should know how to rationalize.

Answer 3

Use the conjugate a pair of situations. a million/(a million + ?3 - ?5) = a million/[(a million + ?3) - ?5] = a million * [(a million + ?3) + ?5] / {[(a million + ?3) - ?5] * [(a million + ?3) + ?5]} = (a million + ?3 + ?5) / [(a million + ?3)² - ?5²] = (a million + ?3 + ?5) / [(4 + 2?3) - 5] = (a million + ?3 + ?5) / (2?3 - a million) = [(a million + ?3 + ?5)(2?3 + a million)] / [(2?3 - a million)(2?3 + a million)] = [(2?3 + 6 + 2?15) + (a million + ?3 + ?5)] / (12 - a million) = (7 + 3?3 + ?5 + 2?15) / 11. (confirmed with Wolfram Alpha.) i'm hoping this facilitates!

Answer 4

The general rule is that you want to multiply and divide by all the possible conjugates of the denominator. In this case, the conjugates are
1+sqrt(3)+sqrt(5),
1-sqrt(3)+sqrt(5), and
1-sqrt(3)-sqrt(5).

The product of these works out to be
-7-3sqrt(3)-sqrt(5)-2sqrt(15).

When this is multiplied by the denominator, the new denominator will be -11. This gives your result.