Question

find under root 13 ans

Answer 1

The underoot is 3.6......

Answer 2

Since 13 is a prime number, there is no simpler form for its square root. √13 is an irrational number somewhere between 3=√9 and 4=√16.

Linearly interpolating, a reasonable first approximation would be:

√13≈3.6=185

We can get better approximations from our initial one (call it a0) using a Newton Raphson method.

A typical formula used to derive a more accurate approximation for √n would be:

ai+1=a2i+n2ai

I prefer to separate ai into numerator pi and denominator qi. So ai=piqi and we can iterate using the formulae:

{pi+1=p2i+nq2iqi+1=2piqi

In our example, n=13, p0=18, q0=5 and we find:

{p1=p20+13q20=324+13⋅25=649q1=2p0q0=180

If we stopped here our approximation would be:

√13≈649180=3.60¯5

Let's try one more iteration:

p2=p21+13q21=421201+13⋅32400=842401q2=2p1q1=233640

Stopping here, we have:

√13≈842401233640≈3.60555127547

Using a calculator:

√13≈3.60555127546398929311