# What is the square root of 30?

##### 2 Answers

You can't get an exact answer by hand, but you can estimate it using some tricks.

so it must be between

What you can begin with is assuming that because:

... the square root of

or more specifically,

Actually, it's not a bad bet. The actual square root is about

You can calculate an approximation by hand as shown below...

#### Explanation:

I explained my favourite method (a sort of Newton Raphson method) for approximating square roots of integers in an answer to the following question:

How do you find the square root 28?

Given an integer

Then iterate using the formulas:

#p_(i+1) = p_i^2 + n q_i^2#

#q_(i+1) = 2 p_i q_i#

If the resulting values of

The successive pairs

For our example, let

#p_1 = p_0^2 + n q_0^2 = 11^2 + 30*2^2 = 121 + 120 = 241#

#q_1 = 2 p_0 q_0 = 2 * 11 * 2 = 44#

This would give

#p_2 = p_1^2 + n q_1^2 = 241^2 + 30*44^2 = 58081 + 58080 = 116161#

#q_2 = 2 p_1 q_1 = 2*241*44 = 21208#

This gives