What is the square root of 84?

2 Answers
Apr 25, 2018

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)

Apr 27, 2018

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 ~~ 1/6th 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