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