what is the easiest way to remember the order of operations over a lifetime.
What different rule when working out exponents any feedback ?
It's through practice.
Wikipedia gives a comprehensive overview of the rules of working with exponents: Exponentiation - Wikipedia, the free encyclopedia
And most of these are fairly intuitive. For example $\displaystyle x^{m-n}$ is equivalent to multiplying $\displaystyle x$ by itself m times and then dividing by $\displaystyle x$ n times, resulting in $\displaystyle x^m/x^n$. Or for instance $\displaystyle x^0 = x^{m-m} = x^m/x^m = 1$.
Hello, C++programmerinCali!
How do YOU keep the rules of exponents straight ?
When I was first learning these Rules, I came up with a "picture" of what was happening.
It was quite primitive (and even childish), but it worked for me.
Think of the "heirarchy" of some basic operations.
First we learned to Count objects: .$\displaystyle 1,2,3\:\hdots$
Then came Addition, which is "repeated counting."
. . (and Subtraction is "repeated take-away.")
Then came Multiplication, which is "repeated addition."
. . (and Division is "repeated subtraction.")
Finally, we learned Powers, which is "repeated multiplication."
. . (and Roots are "repeated division.")
So, written in order, we have:
. . $\displaystyle \text{Counting } \Rightarrow \begin{Bmatrix}\text{ Addition }\\ \text{ Subtraction}\end{Bmatrix} \Rightarrow \begin{Bmatrix}\text{ Multiplication} \\ \text{Division } \end{Bmatrix} \Rightarrow \begin{Bmatrix}\text{ Powers } \\ \text{ Roots} \end{Bmatrix} $
With the Rules of Exponents, we reverse the order . . .
. . $\displaystyle \text{Powers } \underbrace{\Longrightarrow}_{(a^m)^n = a^{mn}} \text{ Multiplication } \underbrace{\Longrightarrow}_{a^m\cdot a^n = a^{m+n}} \text{ Addition} $
. . $\displaystyle \text{Roots } \underbrace{\Longrightarrow}_{(a^m)^{\frac{1}{n}} = a^{\frac{m}{n}}} \text{ Division } \underbrace{\Longrightarrow}_{\frac{a^m}{a^n} = a^{m-n}} \text{ Subtraction}$
I really didn't visualize these diagrams.
I noted that, for exponents, we "step down" an operation.
(For example, to multiply, we add exponents . . . and so on.)