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 $~~ 6.2402$ .