Question

square root of 7 by estimation
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Answer 1

Step-by-step explanation:

7 is between 4 and 9 (2 squared and 3 squared, respectively). Given that it is a bit closer to 9 than 4, I will estimate it as 2.65. In using a calculator, it is approximately 2.646, rounded to the nearest thousandth.

Answer 2

Answer:

2.645751311

Step-by-step explanation:

Since 7 is a prime number, it has no square factors and its square root cannot be simplified.

It is an irrational number, so cannot be exactly represented by $\frac{p}{q}$ for any integers p,q.

We can however find good rational approximations to$\sqrt{7}$.

First note that:

$8^{2} =64=63+1=7*3^{2} +1$

This is in Pell's equation form:

$p^{2} =nq^{2} +1$

with n=7, p=8 and q=3.

This means that $\frac{8}{3}$  is an economical approximation for $\sqrt{7}$ and it also means that we can use $\frac{8}{3}$ to derive the continued fraction expansion of$\sqrt{7}$

$\frac{8}{3} =2+\frac{1}{1\frac{1}{1+\frac{1}{1} } }$

and hence we can deduce:

$\sqrt{7}$≈[2;1,1,1,4]=2+1, 1+1, 1+1, 1+ 4+1, 1+1,1+1, 1+1, 4+1=$\frac{127}{48}$ =2.64583

This is also a solution of Pell's equation for 7, since we find:

$127^{2}=16129=16128+1=7*48^{2}+1$

If you want more accuracy, truncate just before the next 4 or the one after.

By expanding the repeating part of the continued fraction for $\sqrt{7}$ we can derive a generalised continued fraction: and using a calculator, we find:

$\sqrt{7}$≈2.645751311