Video Transcript
In this video, we will learn have
to simplify monomials with an exponent of zero. We will begin by recalling the
quotient rule for exponents, which will be very useful when dealing with expressions
raised to an exponent of zero.
The quotient rule of exponents
states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to
the power of 𝑎 minus 𝑏. When dividing two terms with the
same base, we can subtract the exponents or indices. Let’s consider the expression two
to the third power or two cubed divided by two cubed. Using the quotient rule of
exponents, this simplifies to two to the power of three minus three. As three minus three is equal to
zero, this is equal to two to the zero power or two to the power of zero. Let’s consider what this actually
means.
We know that two cubed is equal to
eight, so we’re dividing eight by eight. Dividing any number or term by
itself gives us an answer of one. As these two things must be
equivalent, two to the power of zero is equal to one. We can generalize this by dividing
𝑥 to the power of 𝑎 by 𝑥 to the power of 𝑎. This simplifies to 𝑥 to the power
of zero as 𝑎 minus 𝑎 is zero. As we are dividing 𝑥 to the power
of 𝑎 by itself, this is also equal to one. This leads us to the general rule
that for any nonzero variable 𝑥, 𝑥 to the power of zero is equal to one. We will now use this rule together
with the quotient rule of exponents to solve some problems.
Determine the value of 12𝑎 to the
power of zero, given that 𝑎 is not equal to zero.
We can begin here but distributing
the power over the monomial. This gives us 12 to the power of
zero multiplied by 𝑎 to the power of zero. We can do this as we know that 𝑥𝑦
raised to the power of 𝑛 is equal to 𝑥 to the power of 𝑛 multiplied by 𝑦 to the
power of 𝑛. We also recall that for any nonzero
𝑥, 𝑥 to the power of zero is equal to one. This means that 12 to the power of
zero is equal to one. As we are told the variable 𝑎
cannot be equal to zero, then 𝑎 to the power of zero is also equal to one. Our expression simplifies to one
multiplied by one, which is equal to one. As any monomial raised to the power
of zero is equal to one, then 12𝑎 to the power of zero is one.
In our next question, we need to
decide whether a statement is true or false.
True or false: 24𝑥 to the power of
zero is equal to 24.
We recall that for any nonzero
value of 𝑥, 𝑥 to the power of zero is equal to one. In this question, we’re multiplying
24 by 𝑥 to the power of zero. This is the same as 24 multiplied
by one. As any number multiplied by one is
itself, 24𝑥 to the power of zero is equal to 24. This means that the equation is
true.
In our next question, we need to
simplify an expression raised to the power of zero.
Simplify 13𝑥 to the power of
zero.
In this question, it is important
to note that only the variable 𝑥 is being raised to the power of zero. We’re multiplying 13 or 13 to the
power of one by 𝑥 to the power of zero. We know that 𝑥 to the power of
zero is equal to one. As 13 to the power of one is just
13, we’re left with 13 multiplied by one. The simplified version of 13𝑥 to
the power of zero is 13.
We will now look at a more
complicated expression and decide whether it is true or false.
True or false: 𝑥 plus 𝑦 to the
power of zero is equal to 𝑥 to the power of zero plus 𝑦 to the power of zero,
where 𝑥 plus 𝑦 is not equal to zero, 𝑥 is not equal to zero, and 𝑦 is not equal
to zero.
We know that for any nonzero 𝑥, 𝑥
to the power of zero is equal to one. We are told that 𝑥 plus 𝑦 is not
equal to zero. This means that raising this to the
power of zero will give us an answer of one. On the right-hand side of the
equation, we have 𝑥 to the power of zero plus 𝑦 to the power of zero. Both of these terms will be equal
to one. This means that 𝑥 to the power of
zero plus 𝑦 to the power of zero is equal to two. As the left-hand side is not equal
to the right-hand side, the statement 𝑥 plus 𝑦 to the power of zero equals 𝑥 to
the power of zero plus 𝑦 to the power of zero is false.
In our final question, we will find
the value of an expression involving zero exponents.
Evaluate negative two to the power
of zero plus 𝐶 to the power of zero, where 𝐶 is a constant.
We know that when we raise any
constant to the power of zero, we get an answer of one. This means that negative two to the
power of zero is equal to one, and 𝐶 to the power of zero is also equal to one. Negative two to the power of zero
plus 𝐶 to the power of zero is, therefore, equal to one plus one. The value of the expression is,
therefore, equal to two.
We will now summarize the key
points from this video. We found out in this video that for
any nonzero variable 𝑥, 𝑥 to the power of zero is equal to one. The quotient rule of exponents
tells us that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥
to the power of 𝑎 minus 𝑏. In particular, this tells us that
𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑎 is equal to 𝑥 to the power
of 𝑎 minus 𝑎. This simplifies to 𝑥 to the power
of zero, which is equal to one.
A common mistake when dealing with
zero exponents is confusing raising to the power of zero with multiplying by
zero. For example, seven to the power of
zero is equal to one, whereas seven multiplied by zero is equal to zero. Raising a number to the power of
zero always gives us an answer of one, whereas multiplying a number by zero is equal
to zero.
