In mathematics, what is the process for multiplying, dividing, adding and subtracting surds with different bases and how do you simplify them?

Step-by-step explanation:

I have never heard the term SURDS before, neat.

Multiplying and dividing are easy, you jut multiply/ divide

and

Adding and subtracting don't always work.  For instance can't be added.  The only time you can is if you simplify them, and doing so gives the same bases.  Now for simplifying.

To simplify you take the bases and just find the prime factorization of it.  or in other words break it down into its prime parts.  let's take 60.  the prime factorization is 2*2*3*5.  Notice there is a 2*2 which is 2 squared.  You can pull this out of a surd.

So after finding the prime factorization, take any prime numbers there are 2 of and pull it out.  You may need to do it multiple times.  for instance if it was something like 2*2*3*3*5 or 2*2*2*2*5 you would pull the double of 2 out AND the double of 3 out in the first one, and in the second there are two doubles of 2, so you would have to take the doubles of 2 out twice, or realize it is the same as 4*4.

Let me know if this did not answer your question.