# How do you find the simplest radical form of 433?

##### 1 Answer

Hmmm...

#### Explanation:

If you mean the simplest form of

If you mean the simplest expression involving a radical with value

**Approximations**

Since

Note that:

#20^2 = 400 < 433 < 441 = 21^2#

So a reasonable approximation is somewhere between

Then:

#433 - (104/5)^2 = 9/25#

Now:

#sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+...)))#

So:

#sqrt(433) = 104/5+(9/25)/(208/5+(9/25)/(208/5+(9/25)/(208/5+...)))#

#color(white)(sqrt(104)) = 1/5(104+9/(208+9/(208+9/(208+...))))#

Hence, if we define a sequence recursively by:

#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 208a_(n+1)+9a_n) :}#

then the ratio between successive terms rapidly converges to

The first few terms are:

#0, 1, 208, 43273, 9002656#

So:

#5sqrt(433) ~~ 9002656/43273-104 = 4502264/43273#

So:

#sqrt(433) ~~ 4502264/(5 * 43273) = 4502264/216365 ~~ 20.808652046#