A good way to remember this is to look at an example of why it works.

x^2 = xx

x^3 = xxx

(x^3)^2 = (xxx)(xxx) = x^6 = x^(2*3) = x^(3+3)

(x^2)^3 = (xx)(xx)(xx) = x^6 = x^(3*2) = x^(2+2+2)

(x^2)(x^3) = (xx)(xxx) = x^5 = x^(2+3)

You start with a bunch of x's multiplied together, and you count them. If you multiply by another bunch of x's (maybe a different length), you make a longer string of x's, and counting them is the same as adding the two original numbers.

Taking something to a power means repeatedly multiplying the same thing. So you put a bunch of identical strings of x's together, and since multiplication is repeated addition and the exponents add (repeatedly in this case), it is the same as multiplying the exponents together.

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"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton