Simplify : 7 root 3 / root 10 + root 3 - 2 root 5 / root 6 + root 5 - 3 root 2 / root 15 + 3 root 2

The answer is

-9√30 + 30√5 + 30√3 + 90√2

We first simplify all of the denominators by multiplying top and bottom by the radical:
(7√3/√10)(√10/√10) + √3 - (2√5/√6)(√6/√6) + √5 - (3√2/√15)(√15/√15) + 3√2

= 7√30/10 + √3 - 2√30/6 + √5 - 3√30/15 + 3√2

Multiply everything by 30 to simplify the fractions:
(7√30/10)(30) + 30√3 - (2√30/6)(30) + 30√5 - (3√30/15)(30) + 30*3√2
=210√30/10 + 30√3 - 60√30/6 + 30√5 - 90√30/15 + 90√2

Dividing we get
7√30 + 30√3 - 10√30 + 30√5 - 6√30 + 90√2

Combining like radicals we have
-9√30 + 30√3 + 30√5 + 90√2