# What is the square root of 84?

##### 2 Answers

#### Explanation:

Pull out the pair of twos that are in common and place them on the outside of the radical.

Inside the radical, you are left with

Hence,

#### Explanation:

The square root of

A square root of a number

Note that:

#9^2 = 81 < 84 < 100 = 10^2#

Hence:

#9 < sqrt(84) < 10#

Since

In fact we find:

#55^2 = 3025 = 3024 + 1 = 84 * 6^2 + 1#

Hence

Consider the quadratic whose zeros are

#(x - (55+6sqrt(84)))(x - (55-6sqrt(84))) = x^2-110x+1#

From this we can define a sequence recursively as follows:

#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 110a_(n+1)-a_n) :}#

The first few terms of this sequence are:

#0, 1, 110, 12099, 1330780#

The ratio between successive terms of this sequence tends rapidly towards

So we find:

#sqrt(84) ~~ 1/6(1330780/12099 - 55) = 665335/(6 * 12099) = 665335/72594 ~~ 9.1651513899#