How do you find the square root of 338?

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Answer 1 · 2017-02-17T21:31:54

#sqrt(338) = 13sqrt(2) ~~ 239/13 ~~ 18.385#

Note that:

#338 = 2*169 = 2*13^2#

If #a, b >= 0# then #sqrt(ab) = sqrt(a)sqrt(b)#

So we find:

#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_1 = 1#

#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#