# How do you find the square root of 338?

Feb 17, 2017

$\sqrt{338} = 13 \sqrt{2} \approx \frac{239}{13} \approx 18.385$

#### Explanation:

Note that:

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

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

So we find:

$\sqrt{338} = \sqrt{{13}^{2} \cdot 2} = \sqrt{{13}^{2}} \cdot \sqrt{2} = 13 \sqrt{2}$

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

Consider the sequence defined by:

${a}_{0} = 0$

${a}_{1} = 1$

${a}_{i + 2} = 2 {a}_{i + 1} + {a}_{i}$

The first few terms are:

$0 , 1 , 2 , 5 , 12 , 29 , 70 , 169 , 408 , \ldots$

The ratio between successive pairs of terms tends towards $\sqrt{2} + 1$.

So we can take $169 , 408$ and approximate $\sqrt{2}$ as:

$\sqrt{2} \approx \frac{408}{169} - 1 = \frac{408 - 169}{169} = \frac{239}{169}$

Then:

$13 \sqrt{2} \approx 13 \cdot \frac{239}{169} = \frac{239}{13} \approx 18.385$