Question 407910
We are given the expression {{{(sqrt(10)+2*sqrt(5))/(sqrt(10)+sqrt(5)) }}}. In general when there are radicals {{{(sqrt(a)+sqrt(c))/(sqrt(a)+sqrt(b)) }}} in the denominator of a fraction
we multiply the fraction by {{{(sqrt(a)-sqrt(b))/(sqrt(a)-sqrt(b)) }}}
(example: {{{1/(sqrt(a)+sqrt(b)) }}} =
 {{{ (1/(sqrt(a)+sqrt(b)))*((sqrt(a)-sqrt(b))/(sqrt(a)-sqrt(b))) }}}

So for the given expression we have that


{{{(sqrt(10)+2*sqrt(5))/(sqrt(10)+sqrt(5)) }}}
{{{((sqrt(10)+2*sqrt(5))/(sqrt(10)+sqrt(5)))*((sqrt(10)-sqrt(5))/(sqrt(10)-sqrt(5))) }}}
{{{((sqrt(10)+2*sqrt(5))*(sqrt(10)-sqrt(5)))/(sqrt(10)*sqrt(10)-sqrt(10)*sqrt(5)+sqrt(5)*sqrt(10)-sqrt(5)*sqrt(5)) }}}
{{{((sqrt(10)+2*sqrt(5))*(sqrt(10)-sqrt(5)))/(10-5)) }}}
{{{((sqrt(10)+2*sqrt(5))*(sqrt(10)-sqrt(5)))/(5)) }}}
We will simplify the numerator
{{{(sqrt(10)*sqrt(10)-sqrt(10)sqrt(5)+2*sqrt(5)*sqrt(10)-2*sqrt(5)*sqrt(5))/(5)) }}}
{{{(sqrt(10)*sqrt(5))/(5)) }}}
{{{(sqrt(2*5)*sqrt(5))/(5)) }}}
{{{(sqrt(2)*5)/(5)) }}}

{{{sqrt(2) }}}