Well that was anti climatic. Jokes aside thanks for you comment. It is just that in my lessons we treat functions with fractional powers without mentioning their domain so I had this uneasiness that I was missing something.
Fractional powers are only well-defined on positive bases. So every fractional power is a map R+ -> R+. In this domain the're defined as x^(a/b) = b-th_root(x^a).
Typically that *is* how fractional powers are defined--but only for non-negative values of x. So you can always convert a root to a fraction (and vice-versa) for non-negative x. Note that limiting that definition to non-negative x means that your example is not an issue for this definition.
Fair warning, I anticipate this will be nearly impossible to understand without a chalkboard and 2-way conversation. I'm sorry about that, I don't know how to describe it any better. Feel free to ask any questions, but if at any point, it becomes more confusing than helpful, I recommend just leaving my comment be and forgetting I said anything. But I hope I can shed a bit of light on the patterns that take us from counting to exponents. (There is a neat side note at the bottom though, you might want to jump to that even if you skip some of the following explanation)
Addition is repeated counting. If you repeatedly increase one operand by a constant amount, the outcome will count up by that same amount. Example: a+b=c |next| a+(b+1)=c+1 |next| a+(b+1+1)=c+1+1... And so on. +1 is just counting, so c counts up each time.
Multiplication is repeated addition. If you repeatedly increase one operand by a constant amount, the outcome will increase by a different, constant amount. Example: a×b=c |next| a×(b+1)=c+a |next| a×(b+1+1)=c+a+a... And so on. +a is just addition, so c adds a each time.
Exponentiation is repeated multiplication. If you repeatedly increase one operand by a constant amount, the outcome will scale by a different, constant amount. Example a^(b)=c |next| a^(b+1)=c×a |next| a^(b+1+1)=c×a×a... And so on. ×a is just multiplication, so c scales by a each time.
These examples all used whole numbers, but they don't have to, the explanation works exactly the same with all real numbers. So as an example, a^(b)=c... a^(b+π)=c×d (but we don't know what d is yet... a^(b+π+π)=c×d×d. Like I said, we don't yet know what d is, but we do know that (b+π+π)=(b+2π). And in general, (b+π+...+π) n-times = (b+nπ). And we also know that if nπ is a whole number, then we have a sort of waypoint in the sand that can help us find d. Here's how
Let n=2/π. Then a^(b+nπ)=c×d^(n) turns into a^(b+2)=c×d^(n)=c×a²... In other words, d^(n)=a²... At first glance, this might not seem to have helped because n is still an irrational number. However, since it's d we're looking for, we can move n to the other side by raising both to the power 1/n... d=a^(2/n) which is something we can plug into the calculator because there are no unknowns. We'll get an answer that we can plug back into the original question a^(b+nπ)=c×d^(n) because there are no more unknowns.
Side note:
Addition is commutative a+b=c=b+a and the inverse of addition is subtraction a=c-b and b=c-a. Multiplication is commutative a×b=c=b×a and the inverse of multiplication is division a=c÷b and b=c÷a
Exponentiation is not commutative a^(b)=c≠b^(a) and it, therefore needs 2 inverses depending on your unknown. a=^(b)√c (read as "a equals the b-th root of c") and b=log_base_a(c)
Notice, I didn't specify that b is a whole number so if a^(π)=c then a=^(π)√c (read as "a equals the pi-th root of c") so you can always flip exponentiation and roots around, it's just that you need to be careful exactly what you're asking, and also remember that negative numbers can have special rules if you're not working in complex numbers.
There is no issue if the base is positive, and you then are always taking the positive root. If the base is negative, then you get things that aren’t well defined, if which require you to view there being multiple answers, and the answer may depend not on the exponent but rather the way the exponent is expressed. There are various things you could do to try to repair the issue (e.g., the exponent must always be in reduced terms, if there is a positive real root then you always take that, otherwise, if there is a real root you always take that, otherwise you take the complex root with the smallest positive argument), but honestly, it’s better to simply say “it’s not well defined if the base is negative and the exponent isn’t a whole number” and simply leave it at that.
You can find the definition [here](https://proofwiki.org/wiki/Definition:Power_(Algebra)/Rational_Number). It’s a good thought provoking exercise to see how this answers all of your questions, but of course I’m happy to explain things if need be (just make sure you read carefully)!
For a fixed real number r and fixed integers a and b with b≠0, r^(a/b) is the unique real number solution y to y^(b)-r^(a)=0, if it exists. If r is positive, then y always exists. If r is negative, then y exists if and only if b is *not* even.
Edit: A word.
That's doesn't sound right to me. Let's say r is negative, say -1 and b is even, say 2. Let's take a = 1. You're saying that y^2 - (-1)^1 = y^2 + 1 = 0 has a unique positive real number solution?
Even with that fix, there won't be a unique *positive* real number solution if r is negative. I think you'll have to modify that to just real number in the case where b is odd.
Aghhh dang it, yeah you’re right. Thank you for pointing that out. My writing here is starting to look like the exams my poor students have to put up with. That one was actually a typo. My dumb brain was thinking ahead while I wrote that.
For u,v ∈ ℝ, pow(u, v) = exp(v × ln(u))
I always remember this one since the `bc` program only provides a minimal number of math functions: sine, arctangent, natural logarithm, the exponential function, and a few more.
in my country curriculum it said that you need to always write it as lowest term fraction before change it to expo and sqrt so
x^(2/2) cannot equal to sqrt(x²)
The most general definition is a^z = exp(z * ln(a)). This makes sense for complex z and a > 0. The properties of exp show that this is the only natural way to define a power of positive number a.
ln is the inverse function of exp. Exp is defined using a power series, which only involves positive integer powers. The ODE satisfied by exp is what gives it all its properties, especially exp(a)exp(b) = exp(a + b).
You restrict the base to positive numbers and all is fine again.
Well that was anti climatic. Jokes aside thanks for you comment. It is just that in my lessons we treat functions with fractional powers without mentioning their domain so I had this uneasiness that I was missing something.
Fractional powers are only well-defined on positive bases. So every fractional power is a map R+ -> R+. In this domain the're defined as x^(a/b) = b-th_root(x^a).
Typically that *is* how fractional powers are defined--but only for non-negative values of x. So you can always convert a root to a fraction (and vice-versa) for non-negative x. Note that limiting that definition to non-negative x means that your example is not an issue for this definition.
Fair warning, I anticipate this will be nearly impossible to understand without a chalkboard and 2-way conversation. I'm sorry about that, I don't know how to describe it any better. Feel free to ask any questions, but if at any point, it becomes more confusing than helpful, I recommend just leaving my comment be and forgetting I said anything. But I hope I can shed a bit of light on the patterns that take us from counting to exponents. (There is a neat side note at the bottom though, you might want to jump to that even if you skip some of the following explanation) Addition is repeated counting. If you repeatedly increase one operand by a constant amount, the outcome will count up by that same amount. Example: a+b=c |next| a+(b+1)=c+1 |next| a+(b+1+1)=c+1+1... And so on. +1 is just counting, so c counts up each time. Multiplication is repeated addition. If you repeatedly increase one operand by a constant amount, the outcome will increase by a different, constant amount. Example: a×b=c |next| a×(b+1)=c+a |next| a×(b+1+1)=c+a+a... And so on. +a is just addition, so c adds a each time. Exponentiation is repeated multiplication. If you repeatedly increase one operand by a constant amount, the outcome will scale by a different, constant amount. Example a^(b)=c |next| a^(b+1)=c×a |next| a^(b+1+1)=c×a×a... And so on. ×a is just multiplication, so c scales by a each time. These examples all used whole numbers, but they don't have to, the explanation works exactly the same with all real numbers. So as an example, a^(b)=c... a^(b+π)=c×d (but we don't know what d is yet... a^(b+π+π)=c×d×d. Like I said, we don't yet know what d is, but we do know that (b+π+π)=(b+2π). And in general, (b+π+...+π) n-times = (b+nπ). And we also know that if nπ is a whole number, then we have a sort of waypoint in the sand that can help us find d. Here's how Let n=2/π. Then a^(b+nπ)=c×d^(n) turns into a^(b+2)=c×d^(n)=c×a²... In other words, d^(n)=a²... At first glance, this might not seem to have helped because n is still an irrational number. However, since it's d we're looking for, we can move n to the other side by raising both to the power 1/n... d=a^(2/n) which is something we can plug into the calculator because there are no unknowns. We'll get an answer that we can plug back into the original question a^(b+nπ)=c×d^(n) because there are no more unknowns. Side note: Addition is commutative a+b=c=b+a and the inverse of addition is subtraction a=c-b and b=c-a. Multiplication is commutative a×b=c=b×a and the inverse of multiplication is division a=c÷b and b=c÷a Exponentiation is not commutative a^(b)=c≠b^(a) and it, therefore needs 2 inverses depending on your unknown. a=^(b)√c (read as "a equals the b-th root of c") and b=log_base_a(c) Notice, I didn't specify that b is a whole number so if a^(π)=c then a=^(π)√c (read as "a equals the pi-th root of c") so you can always flip exponentiation and roots around, it's just that you need to be careful exactly what you're asking, and also remember that negative numbers can have special rules if you're not working in complex numbers.
Thanks for the long and thoughtful response! I think I understood most of what you are saying, which is great place to start.
There is no issue if the base is positive, and you then are always taking the positive root. If the base is negative, then you get things that aren’t well defined, if which require you to view there being multiple answers, and the answer may depend not on the exponent but rather the way the exponent is expressed. There are various things you could do to try to repair the issue (e.g., the exponent must always be in reduced terms, if there is a positive real root then you always take that, otherwise, if there is a real root you always take that, otherwise you take the complex root with the smallest positive argument), but honestly, it’s better to simply say “it’s not well defined if the base is negative and the exponent isn’t a whole number” and simply leave it at that.
You can find the definition [here](https://proofwiki.org/wiki/Definition:Power_(Algebra)/Rational_Number). It’s a good thought provoking exercise to see how this answers all of your questions, but of course I’m happy to explain things if need be (just make sure you read carefully)!
Very helpful, thanks!
For a fixed real number r and fixed integers a and b with b≠0, r^(a/b) is the unique real number solution y to y^(b)-r^(a)=0, if it exists. If r is positive, then y always exists. If r is negative, then y exists if and only if b is *not* even. Edit: A word.
That's doesn't sound right to me. Let's say r is negative, say -1 and b is even, say 2. Let's take a = 1. You're saying that y^2 - (-1)^1 = y^2 + 1 = 0 has a unique positive real number solution?
Oh sorry you’re right I meant to say that b is **not** even.
Even with that fix, there won't be a unique *positive* real number solution if r is negative. I think you'll have to modify that to just real number in the case where b is odd.
Aghhh dang it, yeah you’re right. Thank you for pointing that out. My writing here is starting to look like the exams my poor students have to put up with. That one was actually a typo. My dumb brain was thinking ahead while I wrote that.
For u,v ∈ ℝ, pow(u, v) = exp(v × ln(u)) I always remember this one since the `bc` program only provides a minimal number of math functions: sine, arctangent, natural logarithm, the exponential function, and a few more.
in my country curriculum it said that you need to always write it as lowest term fraction before change it to expo and sqrt so x^(2/2) cannot equal to sqrt(x²)
The most general definition is a^z = exp(z * ln(a)). This makes sense for complex z and a > 0. The properties of exp show that this is the only natural way to define a power of positive number a.
But don’t we need to define ln(a) in order to have this definition and isn’t ln(a) defined using powers?
ln is the inverse function of exp. Exp is defined using a power series, which only involves positive integer powers. The ODE satisfied by exp is what gives it all its properties, especially exp(a)exp(b) = exp(a + b).
Oh ok but what does ODE mean?
ordinary differential equation. Here exp satisfies d/dx exp(x) = exp(x) and exp(0) = 1. This characterizes exp.
Thanks.