All surds are irrational numbers, but not all irrational numbers are surds; for example, (pi) is irrational but it's not a surd. A surd has to be the simplest representation of a number; for example, sqrt(25)=5 so sqrt(25) is not a surd. sqrt(27) is a surd but the cube root of 27 is not a surd, because its simplest form is the number 3. 1/sqrt(2) is a surd, so surds can be fractions (like sqrt(2/3)=sqrt(2)/sqrt(3)). 1/sqrt(2) is usually rationalised to sqrt(2)/2, but it could be argued that 1/sqrt(2) is a simpler representation. sqrt(x) can also be written x^(1/2) or x^1/2, but the square root symbol is the conventional representation. sqrt(8) is a surd but it can be rationalised as 2sqrt(2). Where a root can be factorised and the root applied to the factor to produce a rational number, then this representation reduces the surd to a simpler form. For example, sqrt(12) is sqrt(4*3)=sqrt(4)*sqrt(3)=2sqrt(3); sqrt(500)=sqrt(100)*sqrt(5)=10sqrt(5).