- [Voiceover] We're asked to simplify the given expression.
And we'll take a look at two methods
for simplifying the expression.
We'll first write the expression
using only positive exponents, and we'll do this
using the negative exponent rule shown here.
a raised to the power of negative m,
which is a fraction, has a denominator of one,
is equal to one over a raised to the power of positive m.
So if the base of a is moved to
the denominator, the exponent changes sign.
So looking back at the given expression,
right now it has a denominator of one.
But because two x squared has a negative exponent,
if we move the base of two x squared
to the denominator, the exponent will change
from negative three to positive three.
So if the given expression is equivalent
to the expression where the numerator has a base
of four x to the fourth raised to the third power,
the denominator would have a base
of two x squared raised to the power
of not negative three, but positive three.
And now we'll clear the parentheses
by applying the power property
of exponents, which is shown here.
When we have powers raised to powers,
we multiply the exponents.
But it's important to remember
four x to the fourth is really
four to the first times x to the fourth,
and two x squared is really two
to the first times x to the second.
So now we'll multiply the exponents.
So we'd have four raised to the power
of one times three, which is three,
x raised to the power of four times three which is 12,
divided by two raised to the power of one times three
which is three, so two to the third,
x raised to the power of two times three, which is six.
And now we'll evaluate four to the third
and two to the third.
Four to the third is equal to 64.
So we have 64 x to the 12th,
divided by two to the third is eight.
So we have divided by eight x to the sixth.
And now we'll simplify the quotient
of the coefficients and the quotient of the x's.
If it's helpful we can write this as 64 divided by eight
times x to the 12th divided by x to the sixth.
Well, 64 divided by eight is equal to eight.
And here we're dividing, and the bases are the same,
so we apply the quotient property of exponents shown here,
where if we're dividing and the bases
are the same, we subtract the exponents.
So x to the 12th divided by x to the sixth
is equal to x raised to the power
of 12 minus six, which is six.
The given expression simplifies to eight x to the sixth.
So this is one method to simplify the given expression.
Let's take a look at a second method.
For the second method, we'll begin
by clearing the parentheses by applying
the power property of exponents.
So again this is four to the first, x to the fourth.
This is two to the first, x to the second.
So here we'll multiply the exponents by three.
Here we'll multiply the exponents by negative three.
So the given expression is equivalent to four
raised to the power of one times three which is three,
x raised to the power of four times three which is 12,
times two raised to the power of one
times negative three, which is negative three,
and x raised to the power of two
times negative three, which is negative six.
So this is really four to the third
times x to the 12th times two to the negative third
times x to the negative sixth.
Let's change the order of the multiplication
and write this as four to the third
times two to the power of negative three
times x raised to the power of 12
times x raised to the power of negative six.
Let's write two to the negative three
using a positive exponent, and therefore
we can write this as four to the third divided by two
raised to the power of positive three.
And then we have times x to the 12th
times x to the negative sixth.
We're multiplying and the base is the same,
and therefore we add the exponents.
This would be x raised to the power of 12 plus negative six.
Well, four to the third we know is equal to 64.
Two to the third is equal to eight.
And here we have x raised to the power
of 12 plus negative six, which is positive six.
And finally 64 divided by eight is equal to eight.
So of course the expression also
simplifies to eight x to the sixth.
So while you'll probably prefer
one of these methods over the other,
it is important to understand why both methods
work to simplify the given expression.
I hope you found this helpful.