# How do you convert square root of 32 into a decimal without using a calculator?

Feb 5, 2015

First of all, we can simplify the problem observing that $32 = {2}^{5}$, and thus $\setminus \sqrt{32} = \setminus \sqrt{{2}^{5}}$.

Now, recall that taking the square root of a number means to consider that number to the power of a half, and so $\setminus \sqrt{{2}^{5}} = {\left({2}^{5}\right)}^{\frac{1}{2}} = {2}^{\frac{5}{2}}$

Since $\frac{5}{2} = 2 + \frac{1}{2}$, remembering that ${a}^{b + c} = {a}^{b} \setminus \cdot {a}^{c}$ we have that
${2}^{\frac{5}{2}} = {2}^{2 + \frac{1}{2}} = {2}^{2} \setminus \cdot {2}^{\frac{1}{2}} = 4 \setminus \sqrt{2}$.

The problem is written in a simple form, since $\setminus \sqrt{2}$ is easier to exteem than $\setminus \sqrt{2}$.

Without using a calculator, the only thing you can do is trying to approximate from above and below, as it follows:

$\setminus \sqrt{2}$ surely is a number between $1$ and $2$, because ${1}^{2} < 2 < {2}^{2}$.

Now you can try to square some decimal (and this can be done by hand), and you will find that $1.4 < \setminus \sqrt{2} < 1.5$ This means that
$4 \setminus \cdot 1.4 < \setminus \sqrt{32} < 4 \setminus \cdot 1.5$

You can keep going and find the second decimal digit, and so on, but of course the calculation will get tougher with each steps without a calculator