How do you simplify `sqrt24*sqrt(80/192)`?

First, rewrite this expression as:

`sqrt(4 * 6)sqrt((16 * 5)/(64 * 3))`

We can then use these rules for radicals to begin simplifying the expression:

`sqrt(color(red)(a)/color(blue)(b)) = sqrt(color(red)(a))/sqrt(color(blue)(b))` and `sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))`

`sqrt(4 * 6)sqrt((16 * 5)/(64 * 3)) =>`

`sqrt(4 * 6)sqrt((16 * 5))/sqrt((64 * 3)) =>`

`sqrt(4)sqrt(6)(sqrt(16)sqrt(5))/(sqrt(64)sqrt(3)) =>`

`2sqrt(6)(4sqrt(5))/(8sqrt(3)) =>`

`(2 * 4/8)sqrt(6)sqrt(5)/sqrt(3) =>`

`(8/8)sqrt(6)sqrt(5)/sqrt(3) =>`

`1sqrt(6)sqrt(5)/sqrt(3) =>`

`sqrt(6)sqrt(5)/sqrt(3)`

We can now use the reverse of the rules above to complete the simplification process:

`sqrt(6)sqrt(5)/sqrt(3) =>`

`(sqrt(6)sqrt(5))/sqrt(3) =>`

`sqrt(6 * 5)/sqrt(3) =>`

`sqrt((6 * 5)/3) =>`

`sqrt(30/3) =>`

`sqrt(10)`

Square roots which are multiplied or divided can be combined into one square root:

`sqrt24 xx sqrt(80/192) = sqrt((24xx80)/192)`

Write each number as the product of its factors, using square numbers where possible.

`sqrt((24xx80)/192) =sqrt((6xx4xx16xx5)/(3xx4xx16)`

Simplify:
`sqrt((cancel6^2xxcancel4xxcancel16xx5)/(cancel3xxcancel4xxcancel16)`

`= sqrt10`

If there were no square numbers, you would simply have used the prime factors which would give the same results