Video Transcript
Simplify negative 12𝑥 raised to
the sixth power 𝑦 raised to the fifth power all divided by six 𝑥 raised to the
fourth power 𝑦.
In this question, we are asked to
simplify the quotient of two expressions involving unknown values of 𝑥 and 𝑦. The first thing we can note about
the given expression is that both the numerator and denominator are monomials, since
they are the product of constants and variables where the variables are raised to
nonnegative integer exponents.
When finding the quotient of
monomials, we want to divide each of the shared factors separately. We can do this directly in the
given expression, or we can rewrite the expression to separate these factors. It is not necessary to rewrite the
expression in this form. However, it can be useful to see
how the division occurs.
We can now evaluate each of these
divisions separately. First, we can calculate that
negative 12 over six is equal to negative two. For the next two quotients, we can
note that we are dividing exponential expressions with the same base. So we can simplify this by
recalling that the quotient rule for exponents tells us that for a nonzero base 𝑏,
𝑏 raised to the power of 𝑚 over 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised
to the power of 𝑚 minus 𝑛. In other words, when taking the
quotient of two exponential expressions with the same base, we can instead raise the
base to the difference in the exponents.
Assuming that 𝑥 and 𝑦 are not
equal to zero and applying this to the final two quotients gives us 𝑥 raised to the
power of six minus four and 𝑦 raised to the power of five minus one,
respectively. Evaluating the expression in each
of the exponents gives us negative two times 𝑥 squared multiplied by 𝑦 to the
fourth power, provided 𝑥 and 𝑦 are nonzero.
