First, we need to rationalize the denominator by multiplying the expression by the appropriate form of #1# to eliminate the radical in the denominator:
#sqrt(8)/sqrt(8) * (sqrt(3) + 3sqrt(5))/(2sqrt(8)) => (sqrt(8)(sqrt(3) + 3sqrt(5)))/(2sqrt(8)sqrt(8)) =>#
#(sqrt(8)sqrt(3) + 3sqrt(8)sqrt(5))/(2 * 8) => (sqrt(8)sqrt(3) + 3sqrt(8)sqrt(5))/16#
Next, we can use this rule for radicals to simplify the numerator:
#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#
#(sqrt(8)sqrt(3) + 3sqrt(8)sqrt(5))/16 => (sqrt(8 * 3) + 3sqrt(8 * 5))/16 =>#
#(sqrt(24) + 3sqrt(40))/16#
We can rewrite this expression as:
#(sqrt(4 * 6) + 3sqrt(4 * 10))/16#
And use this rule for radicals (the opposite of the rule above) to further simplify the expression:
#sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))#
#(sqrt(4 * 6) + 3sqrt(4 * 10))/16 => (sqrt(4)sqrt(6) + 3sqrt(4)sqrt(10))/16 =>#
#(2sqrt(6) + (3 * 2sqrt(10)))/16 => (2(sqrt(6) + 3sqrt(10)))/16 =>#
#(sqrt(6) + 3sqrt(10))/8#
