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# Worked example: rational vs. irrational expressions

Video transcript

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.