Video Transcript
Simplify four 𝑥 to the seventh
power times 𝑦 cubed minus six 𝑥 to the seventh power times 𝑦 to the seventh power
all over two 𝑥 to the fifth power.
In this question, we are given an
expression that we need to simplify. We can start by analyzing the
expression we are given to see that it is the quotient of a polynomial and a
monomial. We can divide monomials by using
the quotient rule for exponents. So, we can evaluate this expression
if we can rewrite it in terms of the quotient of monomials. To do this, we recall that we can
split the division of the numerator over each term separately.
However, we do need to be careful
since we have a difference in the numerator. So we will need to include this
difference in the terms when we split the division. Dividing each term in the numerator
separately by the denominator yields four 𝑥 to the seventh power 𝑦 cubed over two
𝑥 to the fifth power minus six 𝑥 to the seventh power 𝑦 to the seventh power over
two 𝑥 to the fifth power.
We can now simplify each term
separately by dividing the like terms and applying the quotient rule that states
that we can divide two exponential expressions with the same base by subtracting
their powers. Let’s apply this process to each
term individually. In the first term, we can divide
the coefficients to get four over two. Then we can divide the 𝑥-terms by
using the quotient rule. This gives us 𝑥 to the power of
seven minus five. Finally, we need to multiply this
by 𝑦 cubed.
We can follow the same process for
the second term. The quotient of the coefficients is
six over two. We then divide the 𝑥-terms to get
𝑥 to the power of seven minus five. And then we multiply this by the
𝑦-term, 𝑦 to the seventh power. We can then evaluate the
coefficients and the exponents to obtain the expression two 𝑥 squared 𝑦 cubed
minus three 𝑥 squared 𝑦 to the seventh power.
