Worked example: rational vs. irrational expressions

Let's think about whether each of these expressions produce rational or irrational numbers. And just as a reminder, a rational number is one-- so if you have a rational number x, it can be expressed as the ratio of two integers, m and n. And if you have an irrational number, this cannot happen. So let's think about each of these. So 9 is clearly a rational number. You can express 9 as 9/1, 18/2, or 27/3. 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 the 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 an 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 is irrational. Now let's think about 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 could view it as 1/1. And 1 is clearly a rational number. You could write it as 1/1, 2/2, 3/3, really any integer over itself. So this is going to be rational. Now, let's do this last expression right over here. 3 times the principal root of 9. Well, what's the principal root of 9? Well, it's 3. So this is going to be 3 times 3, which is equal to 9. And we've already talked about the fact that 9 can clearly be expressed as the ratio of two integers-- 9/1, 27/3, 45/5, all different ways of expressing 9.