How do you simplify #sqrt85#?

1 Answer
Apr 30, 2016

Answer:

#sqrt(85)# is not simplifiable.

Explanation:

The prime factorisation of #85# is:

#85 = 5 * 17#

This contains no square factors, so #sqrt(85)# is not simplifiable.

If you wish, you can factor it as:

#sqrt(85) = sqrt(5)sqrt(17)#

but I don't think that counts as simplification.

The general idea is that if a radicand contains square factors then it can be simplified. For example:

#sqrt(24) = sqrt(2^2*6) = sqrt(2^2)sqrt(6) = 2sqrt(6)#

If it has no square factors then this kind of simplification is not possible.

#color(white)()#
Random Bonus

Any square root is expressible as a repeating continued fraction. in our example:

#sqrt(85) = [9;bar(4,1,1,4,18)]#

#=9+1/(4+1/(1+1/(1+1/(4+1/(18+1/(4+1/(1+1/(1+1/(4+1/(18+...))))))))))#

You can truncate the continued fraction to give you rational approximations for the square root.

For example:

#sqrt(85) ~~ [9;4] = 9+1/4 = 37/4 = 9.25#

#sqrt(85) ~~ [9;4,1,1,4] = 9+1/(4+1/(1+1/(1+1/(4)))) = 378/41 = 9.bar(21951)#

Actually #sqrt(85) ~~ 9.219544457#