- [Voiceover] In this lesson we will simplify
several expressions that require the use
of the power of a product rule of exponents
which is a times b raised to the power of n
equals a raised to the power n
times b raised to the power of n.
So if a product raised to the power of n
as shown here, we raise both factors of the product
to the power of n.
We will also be using the power rule of exponents
which is a raised to the power of m raised to the power of n
equals a raised to the power of m times n
which means when we have power raised to powers
we multiply the exponents.
Looking at number one we have xy cubed squared,
or xy cubed raised to the second power.
Because we have the exponent of two outside the parentheses
of this product, we raise both factors
of this product to the second power.
This is equal to x
raised to the power of two times y to the third
raised to the power of two.
If it's helpful we can write x as x to the first
and then we can apply the power rule of exponents
to simplify further.
This is equal to x raised to the power of one times two
times y raised to the power of three times two.
One times two equals two, three times two equals six.
The expression simplifies to x squared y to the sixth.
Next we have the base of three raised to the power of a
times z raised to the sixth power.
Because we have the exponent of six outside the parentheses
of this product, we raise each factor inside the parentheses
to the sixth power.
This is equal to three raised to the power of a
raised to the sixth power
times z raised to the power of six.
And again, if it's helpful, we can write z
as z raised to the first power.
In this form we have powers raised to powers,
and therefore, we multiply the exponents.
This is equal to three raised to the power of a times six
times z raised to the power of one times six.
A times six equals six a, one times six equals six,
the expression simplifies to
three raised to the power of six a,
z raised to the power of six.
Number three we have the base of negative four
times b to the fifth squared, raised to the second power.
Because of the exponent of two outside the parentheses,
we square both factors of the product
inside the parentheses.
And therefore, this is equal to negative four squared
or raised to the second power times b to the fifth squared
or raised to the second power.
Simplifying further, the square of negative four
is equal to two factors of negative four.
Negative four times negative four is equal to positive 16.
So we have positive 16 times b raised
to the power of five times two.
Five times two is equal to 10.
The expression simplifies to 16b to the 10th.
Let's look at two more examples.
Notice in number four we have a base
of negative two p raised to the fourth power,
or negative two times p raised to the fourth power.
To simplify we will raise both factors
inside the parentheses to the fourth power,
which means in the denominator we have
negative two raised to the fourth
times p raised to the fourth.
Simplifying, the base of negative two
raised to the fourth power means we have
four factors of negative two which equals positive 16,
and this is just p to the fourth,
and therefore, the expression simplifies
to one over 16p to the fourth.
And now for the last example, we have the base of a
raised to the power of negative three b squared,
all raised to the power of four.
Because the exponent of four is outside the parentheses,
we're going to raise both factors
of this product to the fourth power.
This is equal to a raised to the power
of negative three raised to the fourth
times b squared raised to the fourth.
In this form we have powers raised to powers
and therefore we multiply the exponents.
This is equal to a raised to the power
of negative three times four
times b raised to the power of two times four.
Negative three times four equals negative 12,
two times four equals eight.
This simplifies to a raised to the power of negative 12
b raised to the power of eight.
Now we can't leave this expression with a negative exponent,
so we'll end by applying the negative exponent rule,
which is a raised to the power of negative n
equals one over a raised to the power of positive n.
So because a raised to the power of negative 12
equals one over a raised to the power of positive 12,
the expression simplifies to b raised to the power of eight
divided by a raised to the power of positive 12.
This is our simplified expression.
I hope you found this helpful.