Question 424118
<pre><font face = "consolas" color = "indigo" size = 4><b>
{{{

(sqrt(3)-3/sqrt(7))/
(1+sqrt(3)*expr(3/sqrt(7)))

 }}}

Write that as

{{{

(sqrt(3)/1-3/sqrt(7))/
(1/1+expr(sqrt(3)/1)*expr(3/sqrt(7)))

 }}}

Multiply that by {{{sqrt(7)/sqrt(7)}}} written as

{{{(sqrt(7)/1)/(sqrt(7)/1)}}}


{{{

expr((sqrt(7)/1)/(sqrt(7)/1))

((sqrt(3)/1-3/sqrt(7))/
(1/1+expr(sqrt(3)/1)*expr(3/sqrt(7))))

 }}}

Distribute and get:

{{{

(sqrt(21)-3)/(sqrt(7)+3sqrt(3)) }}}

Rationalize the denominator by multiplying by

{{{(sqrt(7)-3sqrt(3))/(sqrt(7)-3sqrt(3))}}}

{{{

((sqrt(21)-3)/(sqrt(7)+3sqrt(3))) 

((sqrt(7)-3sqrt(3))/(sqrt(7)-3sqrt(3))) }}}

{{{

(sqrt(21*7)-3sqrt(21*3)-3sqrt(7)+9sqrt(3))
/
(7-9*3)  }}}

{{{

(sqrt(3*7*7)-3sqrt(7*3*3)-3sqrt(7)+9sqrt(3))
/
(7-27)  }}}

{{{(sqrt(3*7^2)-3sqrt(7*3^2)-3sqrt(7)+9sqrt(3))
/
(-20)  }}}


{{{

(7sqrt(3)-3*3sqrt(7)-3sqrt(7)+9sqrt(3))
/
(-20)  }}}

{{{

(7sqrt(3)-9sqrt(7)-3sqrt(7)+9sqrt(3))
/
(-20)  }}}

{{{(16sqrt(3)-12sqrt(7))/(-20)}}}

Reverse the terms on the top

{{{(-12sqrt(7)+16sqrt(3))/(-20)}}}

Factor -4 out of the top:

{{{(-4(3sqrt(7)-4sqrt(3)))/(-20)}}}

The -4 cancels into the -20 giving 5 on the bottom:

{{{(3sqrt(7)-4sqrt(3))/5}}}

Edwin</pre>