Video Transcript
Calculate negative two-thirds cubed
times negative nine-eighths.
In this question, we are asked to
calculate an expression involving the cube and product of a rational number. In the order of operations, we
evaluate the exponent first. So let’s begin by recalling how we
evaluate a rational number raised to a positive integer exponent.
We can recall that if 𝑎 over 𝑏 is
a rational number and we raise this to the power of 𝑛 for a positive integer 𝑛,
then we can raise the numerator and denominator to the power of 𝑛 separately. To apply this result, we first need
to recall that we can rewrite negative two-thirds as negative two over three. And we will also rewrite negative
nine-eighths as negative nine over eight. We can now take the cube of the
numerator and denominator separately to obtain negative two cubed over three cubed
multiplied by negative nine over eight.
We now need to evaluate the
cubes. We can do this by recalling that
cubing a number means a product of three lots of that number. So negative two cubed is negative
eight and three cubed is 27. We now have the product of two
fractions. We can recall that we can multiply
fractions by multiplying their numerators and denominators. So 𝑎 over 𝑏 times 𝑐 over 𝑑 is
equal to 𝑎 times 𝑐 over 𝑏 times 𝑑. This gives us negative eight times
negative nine over 27 times eight.
We could now evaluate these
products. However, it is easier to cancel the
shared factors first. We can cancel the shared factor of
eight in the numerator and denominator. It is worth noting that the
numerator will then be negative one times negative nine. We can also cancel the shared
factor of nine in the numerator and denominator. This gives us negative one times
negative one over three times one. Finally, we can evaluate the
products. We have negative one times negative
one is one and three times one is three. So we are left with one-third,
which is our final answer.
