Question 281641
<pre><font size = 4 color = "indigo"><b>

{{{(sqrt(c)-sqrt(d))/(sqrt(c)+sqrt(d))}}}

Put parentheses around the numerator and denominator

{{{((sqrt(c)-sqrt(d)))/((sqrt(c)+sqrt(d)))}}}

Form the conjugate of the denominator. that is,
change the sign of the second term of {{{(sqrt(c)+sqrt(d))}}},
making it {{{red((sqrt(c)-sqrt(d))))}}}

Now multiply top and bottom of

{{{((sqrt(c)-sqrt(d)))/((sqrt(c)+sqrt(d)))}}}

by the conjugate {{{red((sqrt(c)-sqrt(d))))}}}


{{{(( sqrt(c)-sqrt(d) )red(( sqrt(c)-sqrt(d) )))/
  (( sqrt(c)+sqrt(d) )red(( sqrt(c)-sqrt(d) )))      }}}

Use FOIL on the top and the bottom:

{{{
(sqrt(c^2)-sqrt(cd)-sqrt(cd)+sqrt(d^2))
/(sqrt(c^2)-sqrt(cd)+sqrt(cd)-sqrt(d^2)) }}}

{{{
(sqrt(c^2)-sqrt(cd)-sqrt(cd)+sqrt(d^2))
/(sqrt(c^2)-cross(sqrt(cd))+cross(sqrt(cd))-sqrt(d^2)) }}}

{{{
(sqrt(c^2)-sqrt(cd)-sqrt(cd)+sqrt(d^2))
/(sqrt(c^2)-sqrt(d^2)) }}}

Combine the middle two terms on top as {{{-2sqrt(cd)}}}

{{{
(sqrt(c^2)-2sqrt(cd)+sqrt(d^2))
/(sqrt(c^2)-sqrt(d^2)) }}}

Change the {{{sqrt(c^2)}}} to just {{{c}}} and {{{sqrt(d^2)}}} to just {{{d}}}

{{{
(c-2sqrt(cd)+d)
/(c-d) }}}

Edwin</pre>