What is the square root of 145?

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Answer 1 · 2015-10-18T09:48:24

$145 = 5 * 29$ is the product of two primes and has no square factors, so $sqrt(145)$ is not simplifiable.

$sqrt(145) ~~ 12.0416$ is an irrational number whose square is $145$

You can find approximations for $sqrt(145)$ in a number of ways.

My current favourite is using something called continued fractions.

$145 = 144+1 = 12^2 + 1$ is of the form $n^2 + 1$

$sqrt(n^2 + 1) = [n;bar(2n)] = n + 1/(2n+1/(2n+1/(2n+1/(2n+...))))$

So

$sqrt(145) = [12;bar(24)] = 12 + 1/(24+1/(24+1/(24+...)))$

We can get an approximation by just truncating the repeating continued fraction.

For example:

$sqrt(145) ~~ [12;24] = 12 + 1/24 = 12.041dot(6)$