- [Voiceover] In this lesson, we'll look at several examples
of simplifying a quotient or a fraction raised to a power.
Remember, a fraction bar means division.
To simplify the expressions, we'll use
the power of a quotient rule of exponents,
which is shown here.
If we have the fraction of a over b raised to power of n,
this is equal to a raised to power of n
over b raised to power of n.
So if we have a fraction raised to a power,
we raise both the numerator and denominator to that power.
We'll also be using the power rule of exponents,
which is a raised to power of m raised to power of n
equals a raised to power of m times n.
So looking at number one, we have the fraction
of 2 over x to fourth cubed, or raised to the third power.
To simplify, we raise both the numerator and denominator
to the third power.
This is equal to 2 raised to the third power
over, divided by, x to fourth, raised to the third power.
In the numerator, 2 to third
is equal to three factors of 2,
2 times 2 times 2 is equal to 8.
In the denominator we have powers raised to powers,
which means we multiply the exponents.
So the denominator is x raised to power of 4 times 3.
4 times 3 is equal to 12.
The expression simplifies to 8 over x to the twelfth
or 8 divided by x to twelfth.
Number two.
We have the fraction of p to the third over q
raised to the fourth power.
Because we have the exponent of 4 outside the parenthesis,
we raise both the numerator and denominator
to the fourth power.
This is equal to p to the third raised to the fourth power
over, or divided by, q raised to the fourth power.
If it's helpful, you can write this q as q to the first.
We begin so we just have q to fourth here.
In the numerator we have powers raised to powers,
we multiply the exponents.
P cubed, raised to the fourth power
is equal to p raised to power of 3 times 4.
In the denominator, we can write this as q
raised to power of 1 times 4, or just q to fourth.
3 times 4 is equal to 12.
This simplifies to q to twelfth
over, or divided by, q to the fourth.
Notice how we cannot simplify here,
because the bases are not the same.
We can only apply the quotient rule of exponents
when we're dividing and the bases are the same,
which does not apply here.
Number three.
We have negative 1 over y to the third
raised to the twelfth power, and therefore, we raise
both the numerator and denominator to the twelfth power.
This is equal to the base of negative 1
raised to the twelfth power divided by, or over,
y to third raised to the twelfth power.
In the numerator we have the base of negative 1
raised to the twelfth power.
So because we have 12 factors, or an even number of factors,
of negative 1, the numerator simplifies to positive 1.
In the denominator we multiply the exponents,
which will be y raised to the power of 3 times 12.
3 times 12 equals 36.
This simplifies to 1 over y raised to power of 36.
Let's look at two more examples.
Looking at number four inside the parenthesis,
notice how we have a product.
But we also have one factor that has a negative exponent.
Applying the negative of exponent property,
which is a raised to power of negative n
equals 1 over a raised to power of positive n,
let's write this product as a quotient.
Let's write m squared times n to the negative 4
as m squared divided by n to the fourth.
This is still raised to power of 3.
And now we'll raise both the numerator and denominator
to the third power,
which will give us m squared raised to the third power
over n to the fourth to the third power.
Here we have powers raised to powers.
So we multiply the exponents.
This is equal to m raised to power of 2 times 3
over, or divided by, n raised to power of 4 times 3.
2 times 3 equals 6.
4 times 3 equals 12.
The expression simplifies to m to the sixth
over, or divided by, n to the twelfth.
Again this is not to simplify
because the bases are not the same.
Well, then for number of five,
let's begin by writing the expression inside the parenthesis
using positive exponents.
3 raised to power of negative 2
is equal to 1 over 3 squared,
and z raised to power of negative 3
is equal to 1 over z cubed.
So let's write this as in parenthesis
1 over 3 squared z to the third,
and it's just still squared.
Now we'll raise both the numerator and denominator
to the second power.
So we have 1 squared all over denominator
of 3 squared z cubed squared.
In the numerator we have 1 squared which is equal to 1.
In the denominator notice how we're squaring this product,
and therefore, we square each factor in the product.
So we have 3 squared, squared times z cubed, squared.
Now we have powers raised to powers,
so we multiply the exponents.
This is equal to 1 over 3 raised to power of 2 times 2,
which is 4,
times z raised to power of 3 times 2, which is 6.
And for the last step, 3 raised to fourth power
is equal to four factors of 3, which equals 81.
The expression simplifies to 1 over 81 z to the sixth.
I hope you found this helpful.