How do you simplify a surd?

To simplify a surd is to ensure that we cannot express an expression with an integer multiple of a square root of a lower number than present currently. e.g. if I was to simplify sqrt(50), I would note sqrt(50)=sqrt(25)sqrt(2)=5sqrt(2) and thus the surd would be simplified.

A good method to check if you can simplify a surd would be to prime factorize the number inside the square root, and if you can find any prime number to the power of 2 or more in the prime factorisation, then you may simplify further.

e.g. sqrt(600) = sqrt(5^2 * 2^3 * 3) = sqrt(5^2)*sqrt(2^2)*sqrt(2*3) = 10sqrt(6)

Note, prime factorisation ensures that you have simplified the surd as far as possible.

A more informal approach would be to spot the number in the square root as a multiple of a perfect square. e.g to simplify sqrt(600) we note that 600 is trivially a multiple of 100, which is a perfect square, so we say sqrt(600) = sqrt(100)sqrt(6) = 10sqrt(6) and it is clear we cannot simplify this, however note we could not prove this without demonstrating this via prime factorisation.

An effective method of simplifying a surd is by using a factor tree and once the factor tree is completed, multiply all prime factors to simplify.