Question

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

Answer 1

First of all, we can simplify the problem observing that `32=2^5`, and thus `\sqrt{32}=\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 `\sqrt{2^5}=(2^5)^{1/2}=2^{5/2}`

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

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

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

`\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<\sqrt{2}<1.5` This means that
`4\cdot 1.4<\sqrt{32}<4\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