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
