- [Voiceover] Welcome to several examples on
how to simplify an expression using the zero exponent rule.
For all the examples, we will assume
all of the variables are nonzero.
Before we begin, let's review
the quotient rule of exponents,
as well as the zero exponent rule.
The quotient rule of exponents states
x raised to the power of a divided by
x raised to the power of 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 zero exponent rule is
a raised to the power of zero equals one,
again, as long as a doesn't equal zero.
Looking at the first example, we should recognize
this quotient is equal to one
because any nonzero value
divided by itself is equal to one.
But, let's simplify this using the quotient rule
of exponents, as well as the zero exponent rule
to show the quotient is one.
Because we are dividing, and the bases
are the same, we subtract the exponents.
M to the fourth divided by m to the fourth
is equal to m raised to the power of four minus four.
Well, four minus four is equal to zero.
This simplifies to m to the zero,
m to the zero is equal to one.
So, this can be used for one example
or one justification as why
any nonzero value raised to the zero power is equal to one.
We know this quotient is equal to one,
and when applying the quotient rule of exponents,
we get the base raised to the power of zero,
which must also be equal to one.
Looking at number two, notice how we have
y to the third divided by y to the third,
which again, we should recognize will simplify to one.
But, once again, let's simplify this using
the quotient rule of exponents and the zero exponent rule.
Let's write this fraction as a product.
Let's write this as negative two times
y to the third divided by y to the third.
And now, let's find this quotient.
Because the bases are the same,
and we're dividing, we subtract the exponents.
This is equal to negative two times
y raised to the power of three minus three.
Three minus three is equal to zero.
This simplifies to negative two times y to the zero.
Y to the zero is equal to one.
This simplifies to negative two times one,
which equals negative two.
Let's look at two more examples.
Number three looks a lot more complicated,
but notice for all the exponential expressions,
the base is b squared, c to the third.
Let's begin by determining the product in the denominator.
Remember, when multiplying, and the bases
are the same, we add the exponents,
which is the product rule of exponents, shown here.
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, we'll leave the numerator in its form for right now.
And, now, multiplying in the denominator,
the common base is b squared, c to the third.
So, b squared, c to the third raised to the second power
times b squared, c to the third raised to the third power
is equal to b squared, c to the third
raised to the power of two plus three.
Two plus three is equal to five.
This simplifies to b squared, c to the third
raised to the fifth, divided by
b squared, c to the third raised to the fifth.
Again, in this form, we should recognize that
this quotient will simplify to one,
but applying the quotient rule of exponents,
because we are dividing, and the bases
are the same, we subtract the exponents.
This is equal to b squared, c to the third,
raised to the power of five minus five,
which equals b squared, c to the third
raised to the zero power, which equals one.
Looking at number four, notice part
of this quotient will simplify to one.
To emphasize this, let's write this as
six times xy squared raised to the fourth,
divided by xy squared raised to the fourth.
Once again, in this form, we should recognize
this quotient will simplify to one
because any nonzero value divided by itself equals one.
Let's go ahead and apply the quotient rule of exponents
and the zero exponent rule one more time.
This is equal to six times, here we're dividing,
and the bases are the same, so we subtract the exponents.
This quotient is equal to xy squared
raised to the power of four minus four.
Four minus four is equal to zero.
This simplifies to six times xy squared
raised to the power of zero.
Any nonzero value raised to the zero power is equal to one.
This simplifies to six times one, which equals six.
I hope you found this helpful.