Video Transcript
If 𝑎 is equal to three, 𝑏 is
equal to negative two, 𝑐 is equal to four, and 𝑑 is equal to negative one, which
of the following expressions results in a rational number? Option (A) 𝑎 plus 𝑏 minus 𝑐 all
over 𝑏 multiplied by 𝑎 plus 𝑏 plus 𝑐 over two 𝑑 minus 𝑏. Option (B) 𝑎 over 𝑑 plus 𝑐 minus
𝑎 multiplied by 𝑏 over 𝑐. Option (C) root 𝑎 over 𝑎 times
root 𝑐 over 𝑐. Option (D) 𝑎 plus 𝑏 plus 𝑐 all
over 𝑏 plus 𝑎 multiplied by 𝑎 plus 𝑏 plus c over two 𝑑 minus 𝑏. Or is it option (E) 𝑎 plus 𝑏 all
over 𝑏 multiplied by 𝑐 plus 𝑑 over 𝑑?
In this question, we are given five
expressions involving given constants 𝑎, 𝑏, 𝑐, and 𝑑. And we need to determine which
expression results in a rational number. We can start by recalling that a
rational number is the quotient of any two integers where the denominator is
nonzero. Therefore, we can answer this
question by evaluating each of the expressions by substituting the given values of
these constants into the expression and seeing which expression is the quotient of
two integers with a nonzero denominator.
We can simplify this process
slightly by noting that each of the expressions involves a product. And we know that the product of any
two rational numbers is rational by the closure property of the product of rational
numbers. So, we do not need to calculate the
full expression if we have the product of two rational numbers. We can conclude that their product
is rational.
Let’s start by evaluating the
expression in option (A). Substituting the values of 𝑎, 𝑏,
𝑐, and 𝑑 into the expression gives us three plus negative two minus four all over
negative two multiplied by three plus negative two plus four over two times negative
one minus negative two. We could evaluate each expression
in the numerators and denominators. However, it is useful to note that
these expressions are the product and sum of integers.
So by the closure properties, these
are all integers. Therefore, we do not need to
evaluate the numerators since these are already integers, we only need to check that
our final answer has a denominator that is nonzero. However, we can calculate that two
times negative one minus negative two is zero. We cannot divide by zero. So, this expression is not defined
for these values of 𝑎, 𝑏, 𝑐, and 𝑑. And so, it is not a rational
number.
We can apply a similar process to
option (B). We note that both factors in this
expression are the quotients of integers. So, we just need to check that we
are not dividing by zero. Substituting the values of 𝑎, 𝑏,
𝑐, and 𝑑 into the expression gives us the following. We can evaluate the expression in
the first denominator to see that negative one plus four minus three is equal to
zero. So, this expression is
undefined. Hence, it is not a rational
number.
The expression in option (C) gives
us a different problem. If we substitute the values of 𝑎
and 𝑐 into the expression, we obtain root three over three times root four over
four. We can start by noting that four is
a perfect square. So, the square root of four is
equal to two. We can then simplify two over four
to give us one half. However, we cannot write this
expression as the quotient of two integers. The reason for this is that there
is no integer whose square is three. In other words, three is not a
perfect square.
In option (D), we can once again
note that both of the numerators and denominators are integers. So, we only need to check that we
are not dividing by zero. However, we have already seen that
two 𝑑 minus 𝑏 is equal to zero. So, this expression is undefined
and not a rational number.
This leaves us with option (E). We see that we are multiplying two
fractions. And once again, we can note that
these are both the quotients of integers. We can then note that both
denominators are nonzero, since 𝑏 is negative two and 𝑑 is negative one. Therefore, both of the factors in
this expression are rational numbers. So, this is the product of two
rational numbers.
Hence, by the closure property of
the multiplication of rational numbers, we can conclude that the expression in
option (E) results in a rational number.
