Properties of powers

I was told at some numerical methods class that addition is a function. If you take a second look at powers, they are also a function. They are taking a number, another number and producing a third. Much like addition. Plotting graphs of [math]\displaystyle{ a^x }[/math] helps to see powers as functions.

• [math]\displaystyle{ a^m \cdot a^n = a^{m \ + \ n} }[/math]. Why not [math]\displaystyle{ a^{m \cdot n} }[/math]? First because of the property below.
  
  Let's call [math]\displaystyle{ a^m = b }[/math]. Then [math]\displaystyle{ b \cdot a^n = \underbrace{a^n \ + \ a^n \ + \ a^n \ +\ ...}_{b \ \text{times}} = \underbrace{b \ + \ b \ + \ b \ +\ ...}_{a^n \ \text{times}} }[/math]
  
  Let's call [math]\displaystyle{ a^n = c }[/math]. Then [math]\displaystyle{ c \cdot a^m = \underbrace{a^m \ + \ a^m \ + \ a^m \ +\ ...}_{c \ \text{times}} = \underbrace{c \ + \ c \ + \ c \ +\ ...}_{a^m \ \text{times}} }[/math]
  
  Now the next step is not really a formal proof, but did you notice that [math]\displaystyle{ c \cdot b = a^n a^m }[/math]? Therefore [math]\displaystyle{ c \cdot b = \underbrace{c \ + \ c \ + \ c \ + \ ...}_{b \ \text{times}} = \underbrace{b \ + \ b \ + \ b \ + \ ...}_{c \ \text{times}} }[/math]. Which leads to the conclusion that [math]\displaystyle{ a^{m + n} }[/math] means that to multiply a number to some power by the same number to some other power is the same as adding up the powers, because we are adding up the number of times of one power to the number of times of the other power. There is a formal proof here or here.

• [math]\displaystyle{ (a^b)^c = a^{bc} }[/math]. Everybody learns at school that [math]\displaystyle{ 2^3 = 2 \cdot 2 \cdot 2 }[/math] and that [math]\displaystyle{ 2 \times 3 = 2 + 2 + 2 = 3 + 3 }[/math]. To confuse both is very common and I myself lost count of how many times I made this very mistake. Let's make [math]\displaystyle{ a^b = n }[/math]. Then we have [math]\displaystyle{ n^c = \underbrace{a^b \cdot a^b \cdot a^b\ ...}_{c \ \text{times}} }[/math] which is the same thing as [math]\displaystyle{ a^{\overbrace{{(b \ + \ b \ + \ b \ + \ ...)}}^{c \ \text{times}}} }[/math]. Hence [math]\displaystyle{ (a^b)^c = a^{bc} }[/math].

• [math]\displaystyle{ a^{-1}= \frac{1}{a} }[/math]. At school it's more common to not use negative exponents. I don't know why but I suspect that there is a complexity associated to thinking "We all learn that the power 10 means to multiply a number by itself ten times. If the power is -10, what does it mean to multiply by itself a number of times that is less than zero???". With calculus and linear algebra the negative exponents feel more natural because of derivatives, integrals, the inverse matrix, and the rules that come along. Maybe a way to see it more clearly is this: [math]\displaystyle{ a^{-n} = a^{(-1)(n)} = {(a^n)}^{-1} }[/math]

• [math]\displaystyle{ \frac{a^n}{a^m} = a^{n \ - \ m} }[/math]. At school I was told you subtract the exponent below from the other above. Why? Because the rule says so. Let's rewrite it: [math]\displaystyle{ a^n \cdot \frac{1}{a^m} = a^n \cdot a^{-m} = a^{n \ + \ (-m)} }[/math]. Assuming [math]\displaystyle{ a^m \neq 0 }[/math].

• [math]\displaystyle{ (ab)^n = a^n b^n }[/math]. I think that the most common way to teach it at school is to tell that we are "distributing the power". An alternative way is this: [math]\displaystyle{ (ab)^n = \underbrace{ab \cdot ab \cdot ab \ ...}_{n \ \text{times}} }[/math]

• [math]\displaystyle{ \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} }[/math]. Rewrite it: [math]\displaystyle{ \frac{a^{-1}}{b^{-1}} = a^{-1} \cdot b = \frac{b}{a} }[/math].
  
  The other way is [math]\displaystyle{ \frac{1}{\frac{a}{b}} = \frac{b}{a} }[/math].
  
  At school most teachers tell the rule "invert the fraction", but it's better to rely on negative exponents than memorizing that rule.

• [math]\displaystyle{ a^{\frac{1}{2}} = \sqrt{a} }[/math]. It's quite natural to grasp the concept of integer powers. On the other hand, fractions impose a level of abstraction because we count with integers. Either a person is one unit or not, there aren't half persons. Let's take a look at the following property:
  
  [math]\displaystyle{ \sqrt[n]{a^n} = (a^n)^{\frac{1}{n}} = (a^n)^{(n^{-1})} = a^1 }[/math]. Assuming that [math]\displaystyle{ n \neq 0 }[/math].
  
  What if [math]\displaystyle{ a \lt 0 \ ? }[/math] Some programs or calculators may do it wrong. The first operation that takes precedence is to simplify the exponent, which results in 1 in the previous case. However, if the program or calculator converts the fraction to decimal first and/or compute the power first, it's going to end at a complex number.
  
  At school I don't remember any teacher using the negative exponents the way I just did. Rational exponents were always a rule with no proof at all.

• [math]\displaystyle{ a^0 = 1 }[/math]. Under the condition that [math]\displaystyle{ a \neq 0 }[/math]. One way to see it as [math]\displaystyle{ \frac{a^n}{a^n} = a^{n \ - \ n} = 1 }[/math]. We are, in fact, taking a number and dividing it by itself.
  
  Another way is to think in terms of limits. [math]\displaystyle{ a^5, \ a^4, \ a^3, \ ... }[/math]. The limit of this sequence is, intuitively, approaching 1. Take a number and calculate the square root of it. Repeat, take the square root of the resulting number. After a few times the number should be closer to 1 (the reason for that it's because there exists a formula to calculate roots).
  
  A third way is to think on the graph of functions. Every exponential function is going to cross the vertical axis at (0, 1).
  
  Why [math]\displaystyle{ 0^0 = \ ? }[/math]. One of the most intuitive ways to see why it's undefined is to think on any function with a limit that doesn't exist at the origin.

• [math]\displaystyle{ a^x = a^y \iff x = y }[/math], with the base being positive and not equal to 1. This is what we use to solve exponential equations. If we have an equality and both bases are equal, it implies that the exponents are also equal. It's best to view this as an equality between functions and what we are trying to find is where each function intersects the other. Think about this: both bases are equal, then what differs both is that one grows faster than the other. If their respective growth rates are different, then at some point in the plane they intersect each other. We can only make such assumption because the exponential is a function with a rate of change that never changes its sign. To prove this property we need the log:
  
  [math]\displaystyle{ \log(a^x) = \log(a^y) \iff x\log(a) = y\log(a) \iff x\frac{\log(a)}{\log(a)} = y\frac{\log(a)}{\log(a)} \iff x = y }[/math] (we know that log is never equal to zero)

• [math]\displaystyle{ a^x = a^y + a^z }[/math]. If you ever noticed, all exercises at school do not have the [math]\displaystyle{ a^z }[/math] term. There is no property for a sum of bases. We cannot assume [math]\displaystyle{ x = y + z }[/math] because there is no such property. What sometimes is possible is to factor it, such as [math]\displaystyle{ 2^x(2^x + 1) = 2^{2x} + 2^x }[/math]. For this same reason there is no property for [math]\displaystyle{ \log(x + y) }[/math]. The log of a sum has no rule or property because there is no rule or property for a sum of two exponentials.

• Irrational exponents. If they do show up in calculus, it'll most likely be cancelled out somewhere. It's hard to think on the meaning of an irrational exponent. One way to simplify it is to think in terms of numerical methods. Think on a fraction that is close to that irrational number, then keep the error under a certain constrain which is acceptable for what you need to do. Another way is this: [math]\displaystyle{ a^{\pi} = a^3 \cdot b }[/math], where [math]\displaystyle{ b = a^{0.1415...} }[/math]. A more advanced way to see it is to take advantage of log, functions, limits, to get a better view of it. For ex: [math]\displaystyle{ 2 \lt \sqrt{2} \lt 3 }[/math]. From that we know that [math]\displaystyle{ a^2 \lt a^{\sqrt{2}} \lt a^3 }[/math]. Progressing further in our reasoning and we are required to think on functions and limits. This is not learned in calculus.