How do you find the square root of 338?

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Answer 1 · 2017-02-17T21:31:54

$\sqrt{338} = 13 \sqrt{2} \approx \frac{239}{13} \approx 18.385$ $sqrt(338) = 13sqrt(2) ~~ 239/13 ~~ 18.385$

Note that:

$338 = 2 \cdot 169 = 2 \cdot 13^{2}$ $338 = 2*169 = 2*13^2$

If $a, b \geq 0$ $a, b >= 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$ $sqrt(ab) = sqrt(a)sqrt(b)$

So we find:

$\sqrt{338} = \sqrt{13^{2} \cdot 2} = \sqrt{13^{2}} \cdot \sqrt{2} = 13 \sqrt{2}$ $sqrt(338) = sqrt(13^2*2) = sqrt(13^2)*sqrt(2) = 13sqrt(2)$

If you would like a rational approximation, here's one way to calculate one...

Consider the sequence defined by:

$a_{0} = 0$ $a_0 = 0$

$a_{1} = 1$ $a_1 = 1$

$a_{i + 2} = 2 a_{i + 1} + a_{i}$ $a_(i+2) = 2a_(i+1) + a_i$

The first few terms are:

$0, 1, 2, 5, 12, 29, 70, 169, 408,...$

The ratio between successive pairs of terms tends towards $sqrt(2)+1$ .

So we can take $169, 408$ and approximate $sqrt(2)$ as:

$sqrt(2) ~~ 408/169 - 1 = (408-169)/169 = 239/169$

Then:

$13sqrt(2) ~~ 13*239/169 = 239/13 ~~ 18.385$