SOLUTION: What is {{{(5-sqrt(3))/(3sqrt(5)+4sqrt(3))}}}?

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Question 85653: What is
%285-sqrt%283%29%29%2F%283sqrt%285%29%2B4sqrt%283%29%29?
Found 2 solutions by scianci, Edwin McCravy:
Answer by scianci(186) About Me  (Show Source):
You can put this solution on YOUR website!
Multiply the numerator and denominator by the conjugate of 3sqrt%285%29+%2B+4sqrt%283%29, which is 3sqrt%285%29+-+4sqrt%283%29:
%28sqrt%285%29+-+sqrt%283%29%29%2F%283sqrt%285%29+%2B+4sqrt%283%29%29 * %283sqrt%284%29+-+4sqrt%283%29%29%2F%283sqrt%285%29+-+4sqrt%283%29%29
Now, do FOIL for the tops and the bottoms become the difference of the squares:
%283sqrt%2820%29+-+4sqrt%2815%29+-+3sqrt%2812%29+-+4%283%29%29%2F%289%285%29+-+16%283%29%29 =
%286sqrt%285%29+-+4sqrt%2815%29+-+6sqrt%283%29+-+12%29%2F%28-3%29 =
%28-6sqrt%285%29+%2B+4sqrt%2815%29+%2B+6sqrt%283%29+%2B+12%29%2F3

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
%285-sqrt%283%29%29%2F%283sqrt%285%29%2B4sqrt%283%29%29

A. Form the "conjugate surd" of the denominator:
1. Copy it over.
2. Change the sign of the second term only.

So the conjugate surd of denominator 3sqrt%285%29%2B4sqrt%283%29%29 is 3sqrt%285%29-4sqrt%283%29%29

B. Put the conjugate surd over itself, forming a fraction which has numerical value 1.

Putting the conjugate over itself:

%283sqrt%285%29-4sqrt%283%29%29%2F%283sqrt%285%29-4sqrt%283%29%29

C.  Multiply the original fraction by this one. This is permissible 
because it only amounts to multiplying by 1.

%285-sqrt%283%29%29%2F%283sqrt%285%29%2B4sqrt%283%29%29×%283sqrt%285%29-4sqrt%283%29%29%2F%283sqrt%285%29-4sqrt%283%29%29

D. Form into one fraction by indicating the multiplication of the numerators
and the denominators:

  

E. Use FOIL and/or distributive law to multiply out the top and bottom:

Here we use FOIL for both:

 
             
Squaring square roots takes away the radical:



The two middle terms in the bottom cancel out:

%2815sqrt%285%29+-+20sqrt%283%29-3sqrt%2815%29%2B4%283%29%29%2F%289%285%29-16%283%29%29

%2815sqrt%285%29+-+20sqrt%283%29-3sqrt%2815%29%2B12%29%2F%2845-48%29

%2815sqrt%285%29+-+20sqrt%283%29-3sqrt%2815%29%2B12%29%2F%28-3%29

Make into 4 fractions:

 

Simplify the signs:

-15sqrt%285%29%2F3+%2B+20sqrt%283%29%2F3+%2B3sqrt%2815%29%2F3+-12%2F3

Cancel what will cancel:

-5sqrt%285%29+%2B+20sqrt%283%29%2F3+%2Bsqrt%2815%29+-4

You might want to rearrange it so that it begins with a positive term:

20sqrt%283%29%2F3-5sqrt%285%29%2Bsqrt%2815%29-4

Edwin