What is the square root of 256.25 ?

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Answer 1 · 2016-12-17T18:51:26

$sqrt(256*25) = 80$

$sqrt(256.25) ~~ 16.00781$

Let's look at both possibilities:

$color(white)()$
Square root of $256*25$

If there is a typo in the question and the decimal point was meant to signify multiplication then we find:

$sqrt(256*25) = sqrt(16^2*5^2) = sqrt(16^2)*sqrt(5^2) = 16*5 = 80$

$color(white)()$
Square root of $256.25$

If the question is correct as stands, then we cannot simplify the square root, but we can approximate it.

Note that $4*256.25 = 1025 = 32^2+1$

Since this is of the form $n^2+1$ its square root is expressible as very regular continued fraction:

$sqrt(1025) = [32;bar(64)] = 32+1/(64+1/(64+1/(64+1/(64+...))))$

Hence:

$sqrt(256.25) = 1/2 sqrt(1025) = 1/2 [32;bar(64)]$

We can truncate the continued fraction to get approximations.

For example:

$sqrt(256.25) ~~ 1/2 [32,64] = 1/2 (32+1/64) = 16+1/128 = 16.0078125 ~~ 16.00781$