Question

How do you simplify `3sqrt20 + 5sqrt45 + sqrt75`?

Answer 1

`3sqrt20+5sqrt45+sqrt75=21sqrt5+5sqrt3`

Explanation:

A square root can be simplified by looking for a square number that's a factor of the number within a square root. We use the rule `sqrt(ab)=sqrta*sqrtb` to simplify square roots.

Looking at just the first term, we see that we have `sqrt20`. This can be simplified by noticing that `20=4xx5`, and `4=2^2`. Then, we can say that `sqrt(20)=sqrt(4xx5)=sqrt4sqrt5=2sqrt5`.

We can follow this logic for all the square roots:

`3sqrt20+5sqrt45+sqrt75`

`=3sqrt(4xx5)+5sqrt(9xx5)+sqrt(25xx3)`

`=3sqrt4sqrt5+5sqrt9sqrt5+sqrt25sqrt3`

`=3(2)sqrt5+5(3)sqrt5+5sqrt3`

`=6sqrt5+15sqrt5+5sqrt3`

The terms that both contain `sqrt5` can be simplified. This is analogous to doing `6x+15x=21x`:

`=21sqrt5+5sqrt3`

Answer 2

`21 sqrt(5) + 5 sqrt(3)`

Explanation:

`3 sqrt(20) = 3 sqrt(4*5) = 3*2 sqrt(5) = 6 sqrt(5)`
`5 sqrt(45) = 5 sqrt(9*5) = 5*3 sqrt(5) = 15 sqrt(5)`
`sqrt(75) = sqrt(25*3) = 5 sqrt(3)`