What is the square root of 67?

2 Answers
Oct 12, 2016

Answer:

#67# is a prime, and cannot be factored......

Explanation:

.........and thus #67^(1/2)# #=# #+-sqrt67#.

Feb 19, 2017

Answer:

#sqrt(67) ~~ 34313/4192 ~~ 8.185353#

Explanation:

#67# is a prime number, so in particular has no square factors. So its square root is irrational and not simplifiable.

There are several methods you can use to find rational approximations.

Here's a method based on the Babylonian method...

To find the square root of a number #n#, choose an initial approximation #p_0/q_0# where #p_0, q_0# are integers.

Then apply the following formulas repeatedly to get better approximations:

#{ (p_(i+1) = p_i^2+n q_i^2), (q_(i+1) = 2 p_i q_i) :}#

In our example, let #n = 67#, #p_0 = 8# and #q_0 = 1#, since #8^2 = 64# is quite close to #67#. Then:

#{ (p_1 = p_0^2+n q_0^2 = 8^2+67*1^2 = 64+67 = 131), (q_1 = 2 p_0 q_0 = 2*8*1 = 16) :}#

#{ (p_2 = p_1^2 + n q_1^2 = 131^2+67*16^2 = 17161+17152 = 34313), (q_2 = 2 p_1 q_1 = 2*131*16 = 4192) :}#

If we stop here, we get:

#sqrt(67) ~~ 34313/4192 ~~ 8.185353#

which is accurate to #6# decimal places.