### Video Transcript

Which of the following expressions
is rational given π equals one and π equals 34? Option (A) negative 39 over π
minus one, option (B) 39π over π minus 34, option (C) 39π over π minus one, or
option (D) π over π.

In order to answer this question,
letβs start by remembering the definition of a rational number. A rational number is a number that
can be expressed as π over π where π and π are integers and π is not equal to
zero. In the four options here, we can
see that weβve got four fractions that contain numerical values and also the
algebraic terms π and π. However, weβre given numerical
values for π and π. So, weβll take each expression in
turn and plug in these values.

Letβs start with the first
expression in option (A). Weβll plug in the value that π is
equal to one. Weβll still have 39 on the
numerator, and weβll have one subtract one on the denominator. This, of course, simplifies to
negative 39 over zero. You might think that this looks
pretty good as a fraction. Weβve got a number on the numerator
and a number on the denominator. But in fact, if youβve ever tried
to divide a number by zero on your calculator, youβll get an undefined answer. And importantly, if we look at our
definition of a rational number, the π, the value on the denominator, cannot be
equal to zero. So, this value of negative 39 over
zero is not a rational number. And so, we can exclude option
(A).

In the expression in option (B),
weβll need to substitute in the value π equals 34 twice as π occurs twice. Weβll, therefore, have the
calculation 39 times 34 over 34 minus 34. Before we rush to calculate 39
multiplied by 34, you might already notice whatβs going to happen on this
denominator. Once again, weβre going to have a
denominator that has a value of zero. So, we know that this expression
when π is equal to 34 would not be rational.

We can use the same method of
plugging in the π- and π-values into option (C). So, it have 39 times 34 over one
minus one. You may have already noticed that
this denominator will also give a value of zero. So, option (C) is not a rational
number when π is one and π is 34.

The expression in option (D) is π
over π which will be is 34 over one. Letβs check if this fits the
definition of a rational number. We have it as a fraction π over π
where π and π are integers. And 34 and one are integers. And of course, the denominator one
is not equal to zero. Therefore, 34 over one is
rational. And so, our answer is (D). π over π is rational when π
equals one and π equals 34.

Before we finish with this
question, letβs just take a quick look at this expression in option (A). We saw that this expression
negative 39 over π minus one is not rational, but thatβs not always the case. If we had any other value other
than π equals one, for example, π equals two, then weβll work out negative 39 over
two minus one, which would give us a rational value of negative 39 over one. The expressions (A), (B), and (C)
are only irrational because they had a denominator of zero, as they did in fact have
integer values on the numerator and denominator.