If 𝑎 is equal to negative
two-thirds, 𝑏 is equal to one-third, and 𝑐 is equal to negative nine-halves, find
the numerical value of 𝑎 cubed times 𝑏 cubed times 𝑐 cubed.
In this question, we are given
three rational numbers 𝑎, 𝑏, and 𝑐 as fractions. And we are asked to evaluate the
product of the cubes of these three numbers.
To answer this question, we can
start by recalling that we can cube any fraction by cubing its numerator and
denominator separately. So 𝑝 over 𝑞 all cubed is equal to
𝑝 cubed over 𝑞 cubed. We can use this to evaluate each of
the cubes in the expression. We can start by substituting the
values of 𝑎, 𝑏, and 𝑐 into the expression to obtain that 𝑎 cubed 𝑏 cubed 𝑐
cubed is equal to negative two-thirds cubed times one-third cubed times negative
We now want to cube each of 𝑎, 𝑏,
and 𝑐. Let’s start with 𝑎 cubed. We know that negative two-thirds is
the same as negative two over three. So we cube the numerator and
denominator to get negative two cubed over three cubed.
We can then follow the same process
for 𝑏 cubed. We have that one-third cubed is
equal to one cubed over three cubed. We can follow the same process to
find 𝑐 cubed. We see that negative nine-halves is
equal to negative nine over two. So negative nine-halves cubed is
equal to negative nine cubed over two cubed.
We could start evaluating this
expression. However, it is easier to cancel
shared factors in the numerator and denominator. To do this, we can use the laws of
exponents to write negative two cubed as negative one cubed times two cubed and
negative nine cubed as negative one cubed times three cubed times three cubed. This gives us the following
We can now start canceling the
shared factors in the numerator and denominator. First, we can cancel the two shared
factors of three cubed in the numerator and denominator. Next, we can cancel the shared
factor of two cubed in the numerator and denominator. This leaves us with a denominator
of one. Finally, we can calculate that
negative one cubed is negative one. So the expression simplifies to
give us just one.