Question 81042
Help! :-( I am having trouble solving this problem. It is a homework assignment due Tuesday! Thank you so very much!! Please show the steps on how to solve also. Thanks!! Rationalize the denominator: 
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{{{5/(sqrt(6)+sqrt(5))}}}

Form the conjugate of the denominator,
The conjugate of a two-term expression
has the same first term, but the second
term has its sign changed.  So the
conjugate of {{{sqrt(6)+sqrt(5)}}} is
{{{sqrt(6)-sqrt(5)}}}. So put this over
itself like this {{{(sqrt(6)-sqrt(5))/(sqrt(6)-sqrt(5))}}}
which just equals to 1.  Now multiply the 
original expression by that:

{{{5/(sqrt(6)+sqrt(5))}}}{{{(sqrt(6)-sqrt(5))/(sqrt(6)-sqrt(5))}}}

Now put parentheses around every expression with two terms:

{{{5/((sqrt(6)+sqrt(5)))}}}{{{((sqrt(6)-sqrt(5)))/((sqrt(6)-sqrt(5)))}}}

Put it all as just one fraction

{{{   (5(sqrt(6)-sqrt(5))) /(sqrt(6)+sqrt(5)) (sqrt(6)-sqrt(5))) }}}

Distribute out the top, and FOIL out the bottom:

{{{ (5*sqrt(6) - 5*sqrt(5))/(sqrt(6)*sqrt(6)-sqrt(6)*sqrt(5)+sqrt(5)*sqrt(6)-sqrt(5)*sqrt(5))}}}

Notice that {{{sqrt(6)*sqrt(6)=6}}}, {{{sqrt(5)*sqrt(5)=5}}}, {{{sqrt(6)*sqrt(5)=sqrt(30)}}}, and {{{sqrt(5)*sqrt(6)=sqrt(30)}}}


{{{ (5*sqrt(6) - 5*sqrt(5))/(6-sqrt(30)+sqrt(30)-5))}}}

The two radical terms in the bottom cancel out and we just have

{{{ (5*sqrt(6) - 5*sqrt(5))/(6-5))}}}

or just

{{{ (5*sqrt(6) - 5*sqrt(5))/1)}}}

or just

{{{ 5*sqrt(6) - 5*sqrt(5))}}}