# How do you simplify #sqrt42# rounded to the nearest tenths place?

##### 1 Answer

#### Explanation:

If you ask a calculator to find

#sqrt(42) ~~ 6.4807406984#

So this is between

So rounded to the nearest tenths place,

How would you find this without a calculator?

Note that:

#6*6 = 36#

#6*7 = 42#

#7*7 = 49#

So we expect

Looking at the graph of

graph{(y-x^2)(y - (36+13(x-6))) = 0 [5.8, 7.2, 32, 51]}

By the time we get to

In fact we find:

#6.5^2 = 42.25#

Using Newton's method, the error in the approximation will be about:

#0.25 / (2*6.5) = 0.25/13 ~~ 0.02#

This is much smaller than

Another way of looking at this is continued fractions.

We find that any number of the form

#sqrt(n(n+1)) = [n;bar(2,2n)] = n + 1/(2+1/(2n+1/(2+1/(2n+1/(2+...)))))#

In our example

#sqrt(42) = 6+1/(2+1/(12+1/(2+1/(12+1/(2+...)))))#

Truncating this, we can find rational approximations, such as:

#sqrt(42) ~~ [6;2] = 6+1/2 = 6.5#

#sqrt(42) ~~ [6;2,12] = 6+1/(2+1/12) = 6+12/25 = 6.48#