Question

How do you simplify `sqrt6 - sqrt 3`?

Answer 1

`sqrt6-sqrt3=sqrt3(sqrt2-1)`

Explanation:

This is already about as simplified as it could be. However, if you wanted to, this can be factored.

Recall that (for `a,b>0`)

`sqrt(ab)=sqrta*sqrtb`

Therefore we can say that

`sqrt6=sqrt(2*3)=sqrt2*sqrt3`

We can then rewrite the original expression:

`sqrt6-sqrt3=(sqrt2*sqrt3)-sqrt3`

Factor `sqrt3` from both terms:

`(sqrt2*sqrt3)-sqrt3=sqrt3(sqrt2-1)`

I wouldn't go so far as to say that `sqrt3(sqrt2-1)` is a simplification of `sqrt6-sqrt3`, but they are equivalent statements. Using `sqrt3(sqrt2-1)` may be helpful in simplifying more complicated expressions, such as:

`(sqrt6-sqrt3)/(sqrt10-sqrt5)=(sqrt3(sqrt2-1))/(sqrt5(sqrt2-1))=sqrt3/sqrt5=sqrt3/sqrt5(sqrt5/sqrt5)=sqrt15/5`