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

1 Answer
Apr 8, 2015

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.