How do you simplify sqrt(1414)1414?

1 Answer
Jun 3, 2017

sqrt(1414)1414 is already in simplest form.

Explanation:

The prime factorisation of 14141414 is:

1414 = 2*7*1011414=27101

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

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Notes

Should the 14141414 in the question have been 14441444?

If it was, then we would find:

1444 = 2*2*19*19 = 38^21444=221919=382

So:

sqrt(1444) = 381444=38

We also find that:

37^2 = 1369 < 1414 < 1444 = 38^2372=1369<1414<1444=382

So sqrt(1414)1414 is an irrational number somewhere between 3737 and 3838.

If you would like a rational approximation, we can start by linearly interpolating between 3737 and 3838, finding:

sqrt(1414) ~~ 37+(1414-1369)/(1444-1369) = 37+45/75 = 37+3/5 = 188/5141437+1414136914441369=37+4575=37+35=1885

This will be slightly less than sqrt(1414)1414.

We find:

(188/5^2) = 35344/25 = 35350/25-6/25 = 1414-6/25(18852)=3534425=3535025625=1414625

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 a=188/5 and b=6/25 we get:

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 sqrt(1414) is a little closer to:

37.60319135392633134161