How do you simplify sqrt(1414)√1414?
1 Answer
Explanation:
The prime factorisation of
1414 = 2*7*1011414=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 = 38^21444=2⋅2⋅19⋅19=382
So:
sqrt(1444) = 38√1444=38
We also find that:
37^2 = 1369 < 1414 < 1444 = 38^2372=1369<1414<1444=382
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√1414≈37+1414−13691444−1369=37+4575=37+35=1885
This will be slightly less than
We find:
(188/5^2) = 35344/25 = 35350/25-6/25 = 1414-6/25(18852)=3534425=3535025−625=1414−625
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