Video Transcript
Simplify 12𝑎 to the fifth power
times 11𝑎 to the 13th power 𝑏 to the 13th power minus 12𝑎 to the fifth power 𝑏
to the 13th power over two 𝑎 to the seventh power 𝑏 squared.
Remember, this fraction line means
to divide. So essentially, here we’re dividing
a polynomial by a monomial. Our first instinct might be to
distribute the parentheses on our numerator. However, we can save ourselves a
little bit of time. We’re going to begin by looking for
the greatest common factor of 12𝑎 to the fifth power and two 𝑎 to the seventh
power 𝑏 squared.
Now, we’re allowed to do this and
divide by this greatest common factor because we can see that any factor of 12𝑎 to
the fifth power must be a factor of the entire numerator. So let’s find the greatest common
factor of these two terms. Now, we can do this by first
considering the numerical parts, then considering the 𝑎-part and then the
𝑏-part. The greatest common factor of 12
and two is two. Two is the largest number that
divides evenly into 12 and two without leaving a remainder.
Then the greatest common factor of
𝑎 to the fifth power and 𝑎 to the seventh power is 𝑎 to the fifth power. There is no 𝑏 in 12𝑎 to the fifth
power. So we’ve found the greatest common
factor. And before we do anything, we’re
going to divide the numerator and the denominator of our fraction then by two 𝑎 to
the fifth power.
To divide 12𝑎 to the fifth power
by two 𝑎 to the fifth power, we first divide 12 by two. That’s six. Then 𝑎 to the fifth power divided
by 𝑎 to the fifth power is one. So 12𝑎 to the fifth power divided
by two 𝑎 to the fifth power is six. Similarly, we divide two 𝑎 to the
seventh power 𝑏 squared by two 𝑎 to the fifth power. Two divided by two is one. Then, 𝑎 to the seventh power
divided by 𝑎 to the fifth power is 𝑎 squared. 𝑏 squared divided by one is 𝑏
squared. So two 𝑎 to the seventh power 𝑏
squared divided by two 𝑎 to the fifth power is 𝑎 squared 𝑏 squared.
This means we can rewrite our
fraction as six times 11𝑎 to the 13th power 𝑏 to the 13th power minus 12𝑎 to the
fifth power 𝑏 to the 13th power over 𝑎 squared 𝑏 squared. Now, we’re ready to distribute our
parentheses. We do so by multiplying six by the
first term and then six by the second. Six times 11𝑎 to the 13th power 𝑏
to the 13th power is 66𝑎 to the 13th power 𝑏 to the 13th power. And then six times negative 12𝑎 to
the fifth power 𝑏 to the 13th power is negative 72𝑎 to the fifth power 𝑏 to the
13th power. And this is all over 𝑎 squared 𝑏
squared.
Now, there are a number of ways we
can evaluate the division at this point. Since that fraction line actually
means divide, we could use the bus stop or we can reverse the process for adding
fractions. And we can separate this into two
individual fractions. Then, much as we did earlier, we’ll
divide by the greatest common factor of our numerator and denominator. And if we wish, we can do this by
the numerical part, then by the 𝑎-part, then by the 𝑏-part separately.
Let’s make the denominator one 𝑎
squared 𝑏 squared. And then the greatest common factor
of 66 and one is one. So we leave these for now. Then, what about the greatest
common factor of 𝑎 to the 13th power and 𝑎 squared? Well, it’s 𝑎 squared. So we divide both the numerator and
denominator by 𝑎 squared. This leaves 𝑎 to the 11th power on
our numerator and then simply one on our denominator.
Similarly, the greatest common
factor of 𝑏 squared and 𝑏 to the 13th power is 𝑏 squared. So when we divide through by 𝑏
squared, we’re simply left with 𝑏 to the 11th power on our numerator. And so our first term is 66𝑎 to
the 11th power 𝑏 to the 11th power over one, or simply 66𝑎 to the 11th power 𝑏 to
the 11th power.
Let’s repeat this process for our
second fraction. 72 and one have a greatest common
factor of one, so we leave them. Then, the greatest common factor of
𝑎 to the fifth power and 𝑎 squared is 𝑎 squared. Dividing through, and we’re left
with 𝑎 cubed on the numerator. Then, the greatest common factor of
𝑏 squared and 𝑏 to the 13th power is 𝑏 squared. And as before, that leaves us with
𝑏 to the 11th power. And so we see that we’re left with
66𝑎 to the 11th power 𝑏 to the 11th power minus 72𝑎 cubed 𝑏 to the 11th
power.
