I know this is very basic but I’ve never been able to wrap my head around them. Please help. 😀
In: 4
Positive exponents is the number of times you multiply something by itself. So x^1 = x, x^2 = x*x, x^3 = x*x*x, and so on.
Now you notice an interesting pattern : for example, (x^2)*(x*3) = (x*x)*(x*x*x) = x^5. And 5 = 2+3. That works for any exponent : (x^5)*(x^10) = x^15, and for any positive numbers n and m, (x^n)*(x^m) = x^(n+m).
Now you can wonder : why doing that only with positive numbers ? What if I try with zero ? Or negative numbers ? x^0 must verify the equation, for any n : (x^0)*(x^n) = x^(n+0) = x^n. If you multiply something by x^n to obtain x^n, the something must be 1. So x^0 = 1.
Now let’s try with x^(-1). Let’s take an example : (x^-1)*(x^3) = x^(3-1) = x^2. So x^(-1) = (x^2) / (x^3) = 1/x.
And you can go on with every negative exponent : x^(-4) = 1/(x^4) = 1/x/x/x/x. That’s not some arbitrary convention, that’s true because of the fundamental property of exponents : (x^n)*(x^m) = x^(n+m).
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