# How do you simplify #sqrt(1414)#?

##### 1 Answer

#### Explanation:

The prime factorisation of

#1414 = 2*7*101#

This contains no square factors, so the square root is already in simplest form.

**Notes**

Should the

If it was, then we would find:

#1444 = 2*2*19*19 = 38^2#

So:

#sqrt(1444) = 38#

We also find that:

#37^2 = 1369 < 1414 < 1444 = 38^2#

So

If you would like a rational approximation, we can start by linearly interpolating between

#sqrt(1414) ~~ 37+(1414-1369)/(1444-1369) = 37+45/75 = 37+3/5 = 188/5#

This will be slightly less than

We find:

#(188/5^2) = 35344/25 = 35350/25-6/25 = 1414-6/25#

That's not bad, but if we want greater accuracy, we can use a generalised continued fraction based on this approximation.

In general we have:

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

Putting

#sqrt(1414) = 188/5 + (6/25)/(376/5+(6/25)/(376/5+(6/25)/(376/5+...)))#

You can terminate this continued fraction to get rational approximations, such as:

#sqrt(1414) ~~ 188/5 + (6/25)/(376/5+(6/25)/(376/5)) = 13291036/353455 ~~ 37.603191353921#

A calculator tells me that

#37.60319135392633134161#