Question

What is the square root of 84?

Answer 1

`2 sqrt(21)`

Explanation:

`sqrt (2^2(3)(7))`

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 `(3)(7)=21`

Hence,

`sqrt(84) = 2 sqrt (21)`

Answer 2

`sqrt(84) ~~ 665335/72594 ~~ 9.1651513899`

Explanation:

The square root of `84` is an irrational number a little larger than `9`.

A square root of a number `n`, is a number `a` such that `a^2 = n`. Actually every positive number has two square roots, but "the" square root is usually taken to mean the positive one. The two square roots of `84` are denoted `sqrt(84)` and `-sqrt(84)`.

Note that:

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

Hence:

`9 < sqrt(84) < 10`

Since `84` is `3/19`'s of the way between `81` and `100`, we can approximate `sqrt(84)` as `3/19 ~~ 1/6`th of the way between `9` and `10`, about `9 1/6 = 55/6`.

In fact we find:

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

Hence `55/6` is a very efficient approximation to `sqrt(84)`

Consider the quadratic whose zeros are `55+6sqrt(84)` and `55-6sqrt(84)`:

`(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 `55+6sqrt(84)`

So we find:

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