Question

What is the square root of 67?

Answer 1

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

Explanation:

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

Answer 2

`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.