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consider the nth root of a when would it be a surd in the previous video we saw that a third is an irrational root of a positive rational number it means a should be a positive rational number and the result should be irrational some examples of a third are cube root of five or square root of 12 there are a few conditions we need to keep in mind the first condition says that this is a surd if N is a natural number greater than one the order or the index has to be greater than one here as we can see the order is 3 + square root means the order is 2 the second condition says that a has to be positive rational number 5 and 12 are positive and rational and the result of nth root of a should be irrational if we calculate the values of these two numbers we will see that they are non terminating and non recurring which makes them irrational so these are the three important conditions that make the number a surd what about the sixth root of nine is it a third well the order is a natural number greater than one the base is a positive rational number and the result will be an irrational number that makes it a surd what about the fourth root of 16 the order is a natural number greater than one the base is a positive rational number but wait is the result irrational no it equals two since the third condition is not satisfied this is not possible there is another important thing we need to know about thirds search with order equal to two are given a special name they are called quadratics thirds now we come to the last in the most interesting concept in this video consider the number cube root of square root of five is this a third let's think of the conditions the order has to be a natural number greater than one this condition is satisfied but what the base here it's square root of five the base is irrational let's go back and see the conditions the second condition says that the base has to be a positive rational number but in the example we can see the base is irrational so does this mean that this is not a surd let's see we know that the nth root of a can be written in exponential form as a raise to 1 by n this is the power or the exponent and this is the base the exponent becomes the reciprocal of the order so the cube root of root 5 can be written as root 5 the whole raised to 1 by 3 this is the base and 1 by 3 is the exponent can we also write the base in exponential form yes it can be written as 5 raised to 1 by 2 and the whole thing is raised to 1 by 3 wait a moment do you see what we are getting at do you happen to remember the power law the power law says that a raise to M the whole raise to n equals a raise to M multiplied by n if we apply it here this will equal 5 raised to 1 by 6 if you look at it closely it is in the a raise to 1 by n form using this formula we can write it as 6th root of 5 here the order is a natural number greater than 1 and the base 5 is a positive rational number so this is 1/3 which makes this 1/3 - after we modified this a bit we realize that it is indeed a third we will understand this part of the video in much more detail in the coming videos you