Question

$\sqrt{404.41}$
Solve the problem

Answer 1

Use a Newton Raphson type method to find:

√404.41≈20.10994779

Explanation:

404.41=202+2.12, so you might think there's a nice expression for √404.41, but not so.

We can say √404.41=√40441100=√40441√100=√4044110

So the problem reduces to finding the square root of the whole number 40441 then dividing by 10.

What's the prime factorisation of 40441?

Trying each prime in turn, we eventually find:

40441=37⋅1093

So 40441 has no square factors and the square root cannot be simplified.

To find a good approximation:


Use a Newton Raphson type method with an initial approximation of 200 as follows:

n=40441
p0=200
q0=1

Iteration step:

pi+1=p2i+nq2i
qi+i=2piqi

So:

p1=p20+nq20=2002+40441⋅12=80441
q1=2p0q0=2⋅200⋅1=400

p2=804412+40441⋅4002=12941314481
q2=2⋅80441⋅400=64352800

This gives an approximation:

√40441≈1294131448164352800≈201.09947789

Hence √404.41≈20.10994779

Actually √40441≈201.09947787



#Ninja

Answer 2

answer is 20.10 if this query