How do I simplify this?

Split the radicals of fractions into a fraction of radicals:

`color(white)=sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)`

`=sqrt2/sqrt3+sqrt3/sqrt2+2*sqrt1/sqrt24`

Simplify a little:

`=sqrt2/sqrt3+sqrt3/sqrt2+2*1/sqrt24`

`=sqrt2/sqrt3+sqrt3/sqrt2+2/sqrt24`

`=sqrt2/sqrt3+sqrt3/sqrt2+2/(2sqrt6)`

`=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6`

Rationalize the denominator or each fraction:

`=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6color(red)(*sqrt6/sqrt6)`

`=sqrt2/sqrt3+sqrt3/sqrt2+sqrt6/6`

`=sqrt2/sqrt3+sqrt3/sqrt2color(red)(*sqrt2/sqrt2)+sqrt6/6`

`=sqrt2/sqrt3+sqrt6/2+sqrt6/6`

`=sqrt2/sqrt3color(red)(*sqrt3/sqrt3)+sqrt6/2+sqrt6/6`

`=sqrt6/3+sqrt6/2+sqrt6/6`

Get a common denominator and simplify:

`=(2sqrt6)/6+(3sqrt6)/6+sqrt6/6`

`=(2sqrt6+3sqrt6+sqrt6)/6`

`=(6sqrt6)/6`

`=sqrt6`

That's the simplification. You can check this using a calculator:

![https://www.desmos.com/calculator]()

Hope this helped!

Simplify:

`sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)`

Apply rule `sqrt(a/b)=sqrta/sqrtb`.

`sqrt2/sqrt3+sqrt3/sqrt2+(2sqrt1)/sqrt24`

Rationalize the denominator in the first term.

`(sqrt2sqrt3)/(sqrt3sqrt3)+sqrt3/sqrt2+(2sqrt1)/sqrt24`

Apply rule `sqrtasqrta=a`.

`(sqrt2sqrt3)/3+sqrt3/sqrt2+(2sqrt1)/sqrt24`

Apply rule `sqrtasqrtb=sqrt(ab)`.

`sqrt6/3+sqrt3/sqrt2+(2sqrt1)/sqrt24`

Rationalize the denominator in the second term.

`sqrt6/3+(sqrt3sqrt2)/(sqrt2sqrt2)+(2sqrt1)/sqrt24`

Apply rule `sqrtasqrta=a`.

`sqrt6/3+(sqrt3sqrt2)/2+(2sqrt1)/sqrt24`

Apply rule `sqrtasqrtb=sqrtab`.

`sqrt6/3+sqrt6/2+(2sqrt1)/sqrt24`

Rationalize the denominator in the last term.

`sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/(sqrt24sqrt24)`

Apply rule `sqrtasqrta=a`.

`sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/24`

Apply rule `sqrtasqrtb=sqrt(ab)`.

`sqrt6/3+sqrt6/2+(2sqrt24)/24`

Prime factorize `sqrt24`.

`sqrt6/3+sqrt6/2+(2sqrt(2^2xx2xx3))/24`

Apply rule `sqrt(a^2)=a`.

`sqrt6/3+sqrt6/2+(2xx2sqrt6)/24`

Simplify.

`sqrt6/3+sqrt6/2+(4sqrt6)/24`

Simplify `(4sqrt6)/24` to `sqrt6/6`.

`sqrt6/3+sqrt6/2+sqrt6/6`

The least common denominator (LCD) is `6`.

Multiply the first two terms by a fractional form of `1` so that each has a denominator of `6`.

`sqrt6/3xxcolor(magenta)2/color(magenta)2+sqrt6/2xxcolor(teal)3/color(teal)3+sqrt6/6`

Simplify.

`(2sqrt6)/6+(3sqrt6)/6+sqrt6/6`

Combine numerators.

`((2sqrt6+3sqrt6+sqrt6))/6`

Simplify.

`(6sqrt6)/6`

Simplify.

`sqrt6`