How do you find the square root of 1414?

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How do you find the square root of 1414?

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Answer 1 · 2016-06-26T17:39:44

$\sqrt{1414}$ $sqrt(1414)$ cannot be simplified, but you can calculate approximations.

For example:

$\sqrt{1414} \approx \frac{35347}{940} \approx 37.60319$ $sqrt(1414) ~~ 35347/940 ~~ 37.60319$

Explanation:

Factorising $1414$ $1414$ into prime factors, we find:

$1414 = 2 \times 7 \times 101$ $1414 = 2 xx 7 xx 101$

which has no square factors. So its square root has no simpler form.

You can find rational approximations for the square root using a kind of Newton Raphson method.

Given an initial approximation $a_{0}$ $a_0$ for $\sqrt{n}$ $sqrt(n)$ , iterate using the formula:

$a_{i + 1} = \frac{a_{i}^{2} + n}{2 a_{i}}$ $a_(i+1) = (a_i^2+n)/(2a_i)$

In order to make the arithmetic less messy, I prefer to split $a_{i} = \frac{p_{i}}{q_{i}}$ $a_i = p_i/q_i$ and iterate using the formulae:

$p_{i + 1} = p_{i}^{2} + n q_{i}^{2}$ $p_(i+1) = p_i^2+n q_i^2$

$q_{i + 1} = 2 p_{i} q_{i}$ $q_(i+1) = 2p_i q_i$

If the resulting $p_{i + 1}$ $p_(i+1)$ and $q_{i + 1}$ $q_(i+1)$ has a common factor, then divide both by that factor before the next iteration.

In our example $n = 1414$ $n=1414$ .

Note that $37^{2} = 1369$ $37^2 = 1369$ and $38^{2} = 1444$ $38^2 = 1444$

So linearly interpolating, choose:

$a_{0} = 37 + \frac{1414 - 1369}{1444 - 1369} = 37.6 = \frac{188}{5}$ $a_0 = 37+(1414-1369)/(1444-1369) = 37.6 = 188/5$

So $p_{0} = 188$ $p_0=188$ , $q_{0} = 5$ $q_0=5$

Then:

$p_{1} = p_{0}^{2} + n q_{0}^{2} = 188^{2} + 1414 \cdot 5^{2} = 35344 + 35350 = 70694$ $p_1 = p_0^2+n q_0^2 = 188^2+1414*5^2 = 35344+35350 = 70694$

$q_{1} = 2 p_{0} q_{0} = 2 \cdot 188 \cdot 5 = 1880$ $q_1 = 2 p_0 q_0 = 2*188*5 = 1880$

These are both divisible by $2$ $2$ , so do that to get:

$p_{1 a} = \frac{70694}{2} = 35347$ $p_(1a) = 70694/2 = 35347$

$q_{1 a} = \frac{1880}{2} = 940$ $q_(1a) = 1880/2 = 940$

If we stopped at this stage we would have:

$\sqrt{1414} \approx \frac{35347}{940} \approx 37.603191489$ $sqrt(1414) ~~ 35347/940 ~~ 37.603191489$

Let's try another iteration:

$p_{2} = p_{1 a}^{2} + n q_{1 a}^{2} = 35347^{2} + 1414 \cdot 940^{2} = 1249410409 + 1249410400 = 2498820809$ $p_2 = p_(1a)^2 + n q_(1a)^2 = 35347^2+ 1414*940^2 = 1249410409 + 1249410400 = 2498820809$

$q_{2} = 2 p_{1 a} q_{1 a} = 2 \cdot 35347 \cdot 940 = 66452360$ $q_2 = 2 p_(1a) q_(1a) = 2*35347*940 = 66452360$

So:

$\sqrt{1414} \approx \frac{2498820809}{66452360} \approx 37.60319135392633158551$ $sqrt(1414) ~~ 2498820809/66452360 ~~ 37.60319135392633158551$

Actually:

$\sqrt{1414} \approx 37.60319135392633134161$ $sqrt(1414) ~~ 37.60319135392633134161$

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How do you find the square root of 1414?

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