- [Voiceover] We're asked to write each quotient
using a single base, do not simplify further,
and then write all answers with positive exponents.
We will be simplifying these quotients
using both the quotient rule of exponents
as well as the negative rule of exponents.
The quotient rule of exponents
states x raised to the power of a
divided by x raised to the power b
equals x raised to the power of a minus b,
which means when dividing, and the bases are the same,
we subtract the exponents.
And the negative rule of exponents
states a raised to the power of negative n
equals one over a raised to the power of positive n.
And we'll use this first example to better understand
why this negative exponent rule makes sense.
So looking at number one we have y to the second
divided by y to the fifth.
Because the bases are the same,
and we're dividing, we subtract the exponents.
This is equal to y raised to the power of two minus five.
Remember for this difference,
it's always the exponent of the numerator
minus the exponent of the denominator.
Two minus five is equal to negative three,
this simplifies to y raised to the power of negative three.
Now we apply the negative rule of exponents,
because we're asked to express the answers
using positive exponents.
So applying the negative exponent rule,
y raised to the power of negative three
equals one over y raised to the power of positive three.
To better understand why this must simplify
to one over y raised to the power of positive three,
or why y raised to the power of negative three
equals one over y to the power of positive three,
let's look at this quotient again
and expand y to the second and y to the fifth.
So we have y to the second divided by y to the fifth,
we know y to the second is equal to two factors of y,
we know y to the fifth is equal to five factors of y.
We know y is none zero, and any none zero value
divided by itself simplifies to one.
Which means y divided by y simplifies to one here
as well as here, notice by simplifying this way
we can see the quotient simplifies
to one over y raised to the power of positive three.
This is one way to show that y raised the power
of negative three, which we get after applying
the quotient rule of exponents is equal
to one over y raised to the power of positive three.
It should also make sense because we know
we have three more factors of y in the denominator
which is the reason why this simplifies
to one over y raised to the power of three.
Let's look at number two.
We have a quotient with a common base of three x,
and therefore we subtract the exponents.
This quotient is equal to the base of three x
raised to the power of four minus six,
four minus six is equal to negative two.
This simplifies to the base of three x
raised to the power of negative two.
But again we are asked to express the answer
using positive exponents.
The base of three x raised to the power of negative two
is equal to one over the base of three x
raised to the power of positive two
using the negative exponent rule.
And this should make sense looking back
at the original quotient, we have four factors of three x
in the numerator, and six factors of three x
in the denominator.
So if we expanded and simplified, we'd be left
with two factors of three x in the denominator.
Now we will leave this in this form
because we're told to simplify the quotient to a single base
which we have, even though we could multiply this out,
we have two factors of three x,
three x times three x is equal to nine x squared,
this is also equal to one over nine x squared,
but we'll go ahead and leave it in this form.
Now looking at number three, before we simplify the quotient
we need to determine the product in the numerator
because we are multiplying and the bases are the same
we add the exponents.
And this is the product rule of exponents,
which states x raised to the power of a
times x raised to the power of b
equals x raised to the power of a plus b.
So y to the third times y to the second
is equal to y raised to the power of three plus two,
we still have y to the ninth in the denominator.
Two plus three is equal to five,
we have y to the fifth divided by y to the ninth,
and now we're dividing, and the bases are the same,
and therefore we subtract the exponents,
this is equal to y raised to the power of five minus nine,
five minus nine is equal to negative four,
this simplifies to y raised to the power of negative four
which equals one over y raised to the power
of positive four.
And looking back at this quotient here,
the final result should make sense
because we have five factors of y in the numerator,
nine factors of y in the denominator,
and therefore after simplifying
you'd be left with four factors of y in the denominator.
And finally for number four,
we do have a common base of negative two z squared,
which is raised to the third power in the numerator,
and to the seventh power in the denominator.
Because we are dividing and the bases
are the same we subtract the exponents.
This quotient is equal to the base of negative two z squared
raised to the power of three minus seven,
three minus seven equals negative four.
This simplifies to the base of negative two z squared
raised to the power of negative four.
To express a this using positive exponent
we apply the negative exponent rule, which means
this is equal to one over the base of negative two z squared
raised to the power of positive four.
Based upon the directions, we will leave this quotient
in this form, but if we did multiply out the denominator
this would be equal to one over 16 z to the eighth.
I hope you found this helpful.