Question

How do you rationalize the denominator `(5 sqrt 6)/(sqrt 10)`?

Answer 1

If you multiply both the numerator and the denominator by `sqrt(10)`
then the denominator will become a non-radical.

`(5sqrt(6))/sqrt(10)`

`= (5sqrt(6))/(sqrt(10))*(sqrt(10))/(sqrt(10))`

`=(5sqrt(60))/(10)`

which can be simplified as
`(5*(2)sqrt(15))/(10)`

`= sqrt(15)`

Answer 2

Two methods:

Method 1

Multiply by `1` in the form `sqrt 10 / sqrt 10` to get:

`(5sqrt6)/sqrt10 * sqrt10/sqrt10 = (5sqrt 60) /10`

Now simplify `sqrt 60 = sqrt (4*15) = 2 sqrt 15` So we have

`(5sqrt6)/sqrt10 = (5sqrt 60) /10 = (5*2 sqrt 15)/10 = sqrt 15`

Method 2

Notice that 6 and 10 are both divisible by 2, so we can start by simplifying:

`(5sqrt6)/sqrt10=(5sqrt3sqrt2)/(sqrt5sqrt2)=(5sqrt3)/sqrt5`

Method 2a: now multiply by `1` in the form `sqrt5/sqrt5` to get:

`(5sqrt6)/sqrt10= (5sqrt3)/sqrt5*sqrt5/sqrt5 = (5sqrt15)/5 = sqrt15`

Method 2b Reduce `a/sqrta = sqrta` , So `5/sqrt5 = sqrt5`
*
`(5sqrt6)/sqrt10= (5sqrt3)/sqrt5 = ((sqrt5)^2sqrt3)/sqrt5=sqrt5sqrt3= sqrt15`