How do you find the square root of 38.94?

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Answer 1 · 2016-08-05T21:19:54

#=6.24#

#sqrt38.94#
#=6.24#

Answer 2 · 2016-08-05T21:19:59

You find it numerically. The actual square root is #~~ 6.2402#.

The pattern with squares to get #(n+1)^2# is to add the sum of #n# and #n+1# onto #n^2#:

#color(blue)((n+1)^2) = n^2 + 2n + 1#

#= color(blue)(n^2 + [(n) + (n+1)])#

Since #60^2 = 3600#, #61^2 = 60^2 + 60 + 61 = 3721#, and #62^2 = 61^2 + 61 + 62 = 3844#.
Since #3894 - 3844 < n#, #3894# is not a perfect square. Furthermore, since #3894/100 = 38.94#, and #100 = 10^2#, if #3894# has no perfect square, neither does #38.94#. Therefore, it has an irregular decimal answer.
From the above pattern, #63^2 = 62^2 + 62 + 63 = 3969#, and #3894# is #40%# of the way from #3844# to #3969#.
Since #x^2# is approximately linear at high #x#, we can estimate the square root of #38.94# to be about #(62 + 0.4) xx 1/10 = color(blue)(6.24)#.

The actual square root is #~~ 6.2402#.