# How do you find the square root of 26/89?

##### 1 Answer

#### Answer:

#### Explanation:

Note that

So

We can find rational approximations to it.

**Generalised continued fraction method**

First here's a little theory...

Suppose:

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

Then:

#a+b/(a+sqrt(n)) = a+b/(a+color(blue)(a+b/(2a+b/(2a+b/(2a+...))))) = sqrt(n)#

Multiplying both ends by

#a^2+color(red)(cancel(color(black)(asqrt(n))))+b = color(red)(cancel(color(black)(asqrt(n)))) + n#

Subtracting

#b = n-a^2#

So if we want a generalised continued fraction to help us approximate a square root

**Application**

Let

Then:

#b = n-a^2 = 26/89 - 1/4 = (104-89)/356 = 15/356#

So:

#sqrt(26/89) = 1/2+(15/356)/(1+(15/356)/(1+(15/356)/(1+...)))#

We can truncate this to give rational approximations.

For example:

#sqrt(26/89) ~~ 1/2+(15/356)/(1+(15/356)/(1+15/356)) = 74273/137416 ~~ 0.54050#

Having found this, we can see that putting

It gives:

#b = 26/89 - (27/50)^2 = 119/222500#

So:

#sqrt(26/89) = 27/50+(119/222500)/(27/25+(119/222500)/(27/25+(119/222500)/(27/25+...)))#

Just a couple of steps of this give us:

#sqrt(26/89) ~~ 27/50+(119/222500)/(27/25) = 129881/240300 ~~ 0.540495#

which is correct to

This continued fraction expansion will give us approximately