Step-by-step explanation:
you need to be more careful about writing the problem.
is this now
(3×sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))
or is it
3×sqrt(5) + (sqrt(3)/sqrt(5)) - sqrt(3)
or is it even something else ? many combinations are possible. but all of them are totally different things.
I assume here the first option. otherwise it would be too easy.
remember,
(a + b)(a - b) = a² - b²
hey ! that's why we want when dealing with square roots. only squares of square roots and no mixed terms.
so, we know what we need to multiply with the denominator : (sqrt(5) + sqrt(3)). this gives us then 5 - 3 or then 2 as denominator.
and as always, to keep the original value of the fraction unchanged, we need to multiply the numerator by the same expression.
so, we have
(3×sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) × (sqrt(5) + sqrt(3))/(sqrt(5) + sqrt(3)) =
= (3×sqrt(5)+sqrt(3))(sqrt(5)+sqrt(3))/(5 - 3) =
= (3×sqrt(5)+sqrt(3))(sqrt(5)+sqrt(3))/2 =
= (3×sqrt(5)×sqrt(5) + 3×sqrt(5)×sqrt(3) + sqrt(3)×sqrt(5) + sqrt(3)×sqrt(3))/2 =
= (3×5 + 4×sqrt(5)×sqrt(3) + 3)/2 =
= (15 + 4×sqrt(5)×sqrt(3) + 3)/2 =
= (18 + 4×sqrt(5)×sqrt(3))/2 =
= 9 + 2×sqrt(5)×sqrt(3) = 9 + 2×sqrt(15)
FYI
if we truly only need to deal with the fraction
sqrt(3)/sqrt(5)
then we just need to multiply the denominator by sqrt(5) to give us 5 there. and again the numerator needs to be multiplied by the same term :
sqrt(3)/sqrt(5) × sqrt(5)/sqrt(5) =
= (sqrt(3)×sqrt(5))/(sqrt(5)×sqrt(5)) =
= (sqrt(3)×sqrt(5))/5 = sqrt(15)/5
you see how simple that was ?
