Question 93986
{{{ ( sqrt(x) ) / ( 3 * sqrt(x) - sqrt(y) ) }}} Start with the given expression



{{{( ( sqrt(x) ) / ( 3 * sqrt(x) - sqrt(y) ))(( 3 * sqrt(x) + sqrt(y) )/( 3 * sqrt(x) + sqrt(y) )) }}}  Multiply the fraction by {{{( 3 * sqrt(x) + sqrt(y) )/( 3 * sqrt(x) + sqrt(y) )}}} (which is a form of 1). This will rationalize the denominator.



{{{( ( sqrt(x) )( 3 * sqrt(x) + sqrt(y) ) / ( 3 * sqrt(x) - sqrt(y) )( 3 * sqrt(x) + sqrt(y) ))}}} Combine the fractions




{{{( ( sqrt(x) )( 3 * sqrt(x) + sqrt(y) ) / ( (3 * sqrt(x))(3 * sqrt(x)) + 3 * sqrt(x)*sqrt(y) - 3 * sqrt(x)*sqrt(y)- (sqrt(y))(sqrt(y))) )}}} Foil the denominator




{{{( ( sqrt(x) )( 3 * sqrt(x) + sqrt(y) ) / ( 9x + 3 * sqrt(x)*sqrt(y) - 3 * sqrt(x)*sqrt(y)- y) )}}} Multiply {{{(3 * sqrt(x))(3 * sqrt(x))}}} to get {{{9x}}} and  multiply {{{-(sqrt(y))(sqrt(y))}}} to get {{{-y}}}




{{{( ( sqrt(x) )( 3 * sqrt(x) + sqrt(y) ) / ( 9x + cross(3 * sqrt(x)*sqrt(y) - 3 * sqrt(x)*sqrt(y))- y) )}}} Notice the common root terms in the denominator add and cancel to zero



{{{( ( sqrt(x) )( 3 * sqrt(x) + sqrt(y) ) / ( 9x - y) )}}} Simplify




{{{( ( 3 * sqrt(x)*sqrt(x) + sqrt(y)*sqrt(x) ) / ( 9x - y) )}}} Distribute {{{sqrt(x)}}} in the numerator



{{{( ( 3 * x + sqrt(xy) ) / ( 9x - y) )}}} Multiply and combine the root terms




So {{{ ( sqrt(x) ) / ( 3 * sqrt(x) - sqrt(y) ) }}} is equivalent to {{{( ( 3 * x + sqrt(xy) ) / ( 9x - y) )}}}