Question

What is the square root of 45?

Answer 1

Notice how 45 has a perfect square factor.

$\sqrt{45}$

$=\sqrt{9}\sqrt{5}$

$\pm 3\sqrt{5}$

Now if you wanted the decimal answer, you could estimate it.

$\left|\sqrt{4}\right|=2$

$\left|\sqrt{9}\right|=3$

You could say with reasonable accuracy that:

$\left|\sqrt{5}\right|\approx \frac{5-4}{9-4}\cdot (3-2)+2\approx 2.2$

... making $\sqrt{45}\approx \pm 3\cdot 2.2=\pm 6.6$.

In actually, $\left|\sqrt{5}\right|\approx 2.236$ and $\sqrt{45}\approx \pm 6.708$, so it's not too bad of a guess.