Lesson Video: Simplifying Monomials - Zero Exponents | Nagwa Lesson Video: Simplifying Monomials - Zero Exponents | Nagwa

# Lesson Video: Simplifying Monomials - Zero Exponents Mathematics

In this video, we will learn how to simplify monomials with an exponent of zero, as any nonzero number raised to the zero power is equal to one.

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### 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 π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π. When dividing two terms with the same base, we can subtract the exponents or indices. Letβ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. Letβs consider what this actually means.

We know that two cubed is equal to eight, so weβ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 π₯ to the power of π by π₯ to the power of π. This simplifies to π₯ to the power of zero as π minus π is zero. As we are dividing π₯ to the power of π by itself, this is also equal to one. This leads us to the general rule that for any nonzero variable π₯, π₯ 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 12π to the power of zero, given that π 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 π to the power of zero. We can do this as we know that π₯π¦ raised to the power of π is equal to π₯ to the power of π multiplied by π¦ to the power of π. We also recall that for any nonzero π₯, π₯ 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 π cannot be equal to zero, then π 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 12π 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: 24π₯ to the power of zero is equal to 24.

We recall that for any nonzero value of π₯, π₯ to the power of zero is equal to one. In this question, weβre multiplying 24 by π₯ to the power of zero. This is the same as 24 multiplied by one. As any number multiplied by one is itself, 24π₯ 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 13π₯ to the power of zero.

In this question, it is important to note that only the variable π₯ is being raised to the power of zero. Weβre multiplying 13 or 13 to the power of one by π₯ to the power of zero. We know that π₯ to the power of zero is equal to one. As 13 to the power of one is just 13, weβre left with 13 multiplied by one. The simplified version of 13π₯ 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: π₯ plus π¦ to the power of zero is equal to π₯ to the power of zero plus π¦ to the power of zero, where π₯ plus π¦ is not equal to zero, π₯ is not equal to zero, and π¦ is not equal to zero.

We know that for any nonzero π₯, π₯ to the power of zero is equal to one. We are told that π₯ plus π¦ 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 π₯ to the power of zero plus π¦ to the power of zero. Both of these terms will be equal to one. This means that π₯ to the power of zero plus π¦ to the power of zero is equal to two. As the left-hand side is not equal to the right-hand side, the statement π₯ plus π¦ to the power of zero equals π₯ to the power of zero plus π¦ 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 πΆ to the power of zero, where πΆ 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 πΆ to the power of zero is also equal to one. Negative two to the power of zero plus πΆ 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 π₯, π₯ to the power of zero is equal to one. The quotient rule of exponents tells us that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π. In particular, this tells us that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π. This simplifies to π₯ 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.

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