Can someonehelp me rationalize the denominator of this equation?

Question

the main question that I have is how to rationalize this equation (the steps). here is the equation: 1/1+rad3-rad5

the answer should be: 7+3rad3+rad5+2rad15/11.....how do you get this answer?

Answer 1

If you multiply the top and bottom by (1+ rad3) +(rad5) then the denomenator becomes (1+rad3)^2 -(rad5)^2 by the difference of two squares.

Furthermore,

(1+rad3)^2 -(rad5)^2 = 1 +2rad3 +3 - 5

= 2rad3 -1.

Now if you multiple the top and bottom by (2rad3 +1)
you get the fraction:

[(1+ rad3) +(rad5)](2rad3 +1) / [(2rad3)^2 -1^2)]

= [(1+ rad3) +(rad5)](2rad3 +1) / [(4*3 -1)]

=[(1+ rad3) +(rad5)](2rad3 +1) / 11

=[2rad3 +2(rad3)^2 + 2rad15 +1 + rad3 +rad5] / 11

=[2*3 + 1 +3rad3 +rad5 + rad15] / 11

=(7 + 3rad3 +rad5 + rad15) / 11

Answer 2

Multiply numerator and denominator by ( 1 - √3 + √5 ) .

Denominator becomes 2√15 - 7 .

Multiply numerator and denominator by ( 2√15 + 7 ) .

Numerator becomes 7 + 3√3 + √5 + 2√15 .

Denominator becomes 11 .

( Note -2√45 = -6√5 and 2√75 = 10√3 ) .