# How do you find the square root of 50?

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We can also find rational approximations to it.

For example:

#sqrt(50) ~~ 7 14/197 ~~ 7.071066#

#### Explanation:

The square root of *irrational* number, but you can simplify it or find rational approximations for it.

First note that

#50 = 2 xx 5 xx 5#

contains a square factor

#sqrt(50) = sqrt(5^2*2) = sqrt(5^2)*sqrt(2) = 5 sqrt(2)#

Apart from simplifying it algebraically, what is its numerical value?

Note that

In fact, since *continued fraction*:

#sqrt(50) = 7+1/(14+1/(14+1/(14+1/(14+1/(14+1/(14+...))))))#

This can be written as

We can terminate the continued fraction early to give us rational approximations for

For example:

#sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)#

#sqrt(50) ~~ [7;14,14] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066#

In fact:

#sqrt(50) ~~ 7.071067811865475244#

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