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

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 $45 = 9 \cdot 5$.

So $\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3 \sqrt{5}$

That means we can rewrite: $\sqrt{45} + 2 \sqrt{5}$ as
$3 \sqrt{5} + 2 \sqrt{5}$.

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 $\sqrt{5}$ 's

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

$\sqrt{45} + 2 \sqrt{5} = \sqrt{9 \cdot 5} + 2 \sqrt{5} = = 3 \sqrt{5} + 2 \sqrt{5} = 5 \sqrt{5}$

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

If you also want to learn how to get an approximation for $\sqrt{5}$ without a calculator, post that question.