# How do you find the square root of 43?

##### 2 Answers

See below

#### Explanation:

If you are looking at approximation methods that you can employ using pen, paper and some mental arithmetic, you can try a Binomial Expansion.

If you start at 36, a square number, so that you are looking for

You can then use the Binomial Expansion , ie:

In this case:

Even just the first two terms give

We could get a little closer by using a different square number. If you start at 49, another square number, you are now looking at:

Using **just the first 2 terms** of the Binomial Expansion:

And

#### Explanation:

We can find approximations to it as follows...

Note that

So a good first approximation for

We find:

#(13/2)^2 = 169/4 = 42.25#

Given

#sqrt(n) = a+b/(2a+b/(2a+b/(2a+...)))#

where

So in our example:

#n = 43# ,#a = 13/2# and#b=43-169/4 = 3/4#

So:

#sqrt(43) = 13/2+(3/4)/(13+(3/4)/(13+(3/4)/(13+...)))#

We can truncate this to get rational approximations.

For example:

#sqrt(43) ~~ 13/2+(3/4)/13 = 341/52 ~~ 6.5577#

#sqrt(43) ~~ 13/2+(3/4)/(13+(3/4)/13) = 8905/1358 ~~ 6.557437#

A calculator tells me:

#sqrt(43) ~~ 6.5574385243#

See https://socratic.org/s/aCh3Xasm for another example and explanation of this method.