Video Transcript
Three raised to the fourth power is
equal to 81, and three raised to the power of four minus one is equal to 81 divided
by three is equal to 27. Also, three raised to the power of
three minus one is equal to 27 divided by three is equal to nine. By following this pattern, evaluate
three raised to the zeroth power.
In this question, we are given a
pattern involving three raised to increasingly lower exponents. And we are asked to use this
pattern to evaluate three raised to the zeroth power. It is worth noting that we can
directly answer this question by recalling the zero exponent rule, which tells us
any nonzero base raised to the zeroth power is equal to one. This means that we know that the
answer is that three raised to the zeroth power is one. However, this question wants us to
find this value using the given pattern. To do this, let’s take a closer
look at the pattern we are given.
The first line tells us that three
raised to the fourth power is 81. We then divide this equation
through by three and evaluate to obtain 27. We can think of this as a method of
using three raised to an exponent to find the value of three raised to the pervious
exponent. If we know that three raised to the
fourth power is 81, then we can divide this value by three to obtain that three
cubed is 27. We can continue following this
pattern, since we know that three cubed over three is equal to three squared. We can then calculate that 27 over
three is nine. So three squared is nine.
Let’s see what happens if we keep
following this pattern. We can divide the equation through
by three to find three raised to the first power. We can then calculate that nine
divided by three is equal to three. So three raised to the first power
is three. If we apply this process one more
time, then we should find an expression for three raised to the zeroth power. We can calculate that three divided
by three is equal to one. So this pattern also tells us that
three to the zeroth power is equal to one.
There are a few things worth noting
about this pattern. First, we do not need to stop at an
exponent of zero. For instance, we can divide through
by three once more to find three raised to the power of negative one. This gives us that three to the
power of negative one is one divided by three.
Another thing worth noting is that
we can apply this idea to any nonzero base. For example, let 𝑏 be nonzero and
consider the equation 𝑏 raised to the fourth power is equal to 𝑏 times 𝑏 times 𝑏
times 𝑏. Since 𝑏 is nonzero, we can divide
both sides of the equation by 𝑏 to obtain 𝑏 cubed is 𝑏 times 𝑏 times 𝑏. We can then continue this pattern
in the same way we did for 𝑏 equals three. We see that 𝑏 to the zeroth power
should be equal to 𝑏 over 𝑏, which we know is one when 𝑏 is nonzero.
This also shows us why zero raised
to the zeroth power is not defined in this way, since we cannot divide both sides of
these equations by zero. This pattern is our justification
of the zero exponent rule. And it shows us that three raised
to the zeroth power is equal to one.
