Question

How do you simplify `sqrt45 + 2sqrt5` without a calculator?

Answer 1

Can `sqrt 45` be simplified? Is `45` divisible by any perfect squares?
Start at the beginning: `2^2=4` is a perfect square, but `45` is not divisible by `4`.

`3^2=9` is a perfect squares and `45= 9 * 5`.

So `sqrt 45 = sqrt(9*5)=sqrt9 * sqrt5=3sqrt5`

That means we can rewrite: `sqrt45 + 2 sqrt 5` as
`3sqrt5 + 2sqrt5`.

Now, if I have 3 of these things and I add 2 of the same thing, I obviously get 5 of the things. What things? Well, in this case the things are `sqrt5` 's

When you do it all at once, it looks like this:

`sqrt 45 +2sqrt5= sqrt(9*5) + 2sqrt5 ==3sqrt5+2sqrt5=5sqrt5`

In many (most?) algebra classes, this is what we mean by "simplify".

If you also want to learn how to get an approximation for `sqrt5` without a calculator, post that question.