This is part of a series on common misconceptions.
Is this true or false?
Why some people say it's true: A base to the power of is .
Why some people say it's false: An exponent with the base of is .
The statement is ambiguous and has been long debated in mathematics.
This is mostly a matter of definition. Mathematicians love to define things. (After all, how else can we talk about mathematics if we don't know the definitions?)
Many sources consider to be an "indeterminate form," or say that is "undefined." On the other hand, other sources/branches of mathematics define Note that, certainly,
Some of the arguments for why is indeterminate or undefined are as follows:
Argument 1: We know that for all but for all This contradiction means should be left undefined.
Argument 2: With respect to limits, if , then doesn't necessarily tend to any particular value. For example, but Most of the arguments for why defining is useful surround the fact that in some formulas, makes the formula true for special cases involving 0.
Example 1: The binomial theorem says that . In order for this to hold for , we need .
Example 2: The power rule in differentiation states that . In order for this to hold for and , we need .
Example 3: represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1. This is also the same reason why anything else raised to the power of 0 is 1.
Rebuttal: has to be , since , , .
Reply: We cannot generalize from that pattern alone. Certainly, does not equal 0, since we cannot divide by 0.
Rebuttal: has to be since formulas like the binomial theorem would not work when
Reply: Mathematics is a subject built upon definitions--there is no "universal truth" of what really equals. If it is convenient for the binomial theorem to assume that is fine. On the other hand, if a mathematician who works with limits chooses to leave as undefined, that is fine too!
Rebuttal: Why do some problems on Brilliant say that is undefined?
Reply: As explained in this wiki, some sources argue that is undefined. In particular, many algebra courses (prior to the university level) choose to define in this way. Thus, some problem authors--especially in basic algebra problems--may use this definition of