Worked example: rational vs. irrational expressions

let's think about whether each of these expressions produce rational or irrational numbers and this is a reminder a rational number is one so if you have a rational number X it can be it can be expressed as the ratio of two integers m and N and if you have an irrational number this cannot this cannot happen so let's let's think about each of these so 9 is clearly a rational number you can express 9 as 9 over 1 or 18 over 2 or 27 over 3 so it can clearly be expressed as the ratio of two integers but what about the square root of 45 so let's think about that a little bit square root of 45 that's the same thing as the square root of 9 times 5 which is the same thing as the square root of 9 times the square root of 5 the principal root of 9 is 3 so it's 3 times the square root of 5 so this is going to be 9 plus 3 times the square root of 5 so square root of 5 is irrational you're taking the square root of a non perfect square right over here irrational 3 is rational but the product of a rational and an irrational is still going to be irrational so that's going to be irrational and then you're taking a irrational number and you're adding 9 to it you're adding a rational number to it but you add a rational to an irrational and you're still going to have an irrational so this whole thing this whole thing is irrational irrational now let's think about now let's think about this this expression right over here well the numerator can be rewritten as the square root of 9 times 5 over 3 times the square root of 5 well that's the same thing as the square root of 9 times the square root of 5 over 3 times the square root of 5 well that's the same thing as 3 times the square root of 5 over 3 times the square root of 5 well that's just going to be equal to 1 or you can view this 1 over 1 and one is clearly a rational number you can write it as 1 over 1 2 over 2 3 over 3 really any integer over itself so this is going to be rational now let's do this last expression right over here times the principal root of nine but what's the principal root of nine plus three so this is going to be 3 times 3 which is equal to 9 and we've already talked about the fact that mine can clearly be expressed as the ratio of two integers 9 over 1 27 over 3 45 over 5 all different forms of different ways of expressing 9