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Explanation:
It starts with the basic idea of an exponent being a notation for the number of times the base appears as a factor.
For example, x^2 = x · x . . . . . the base x appears as a factor 2 times.
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Then the rules for multiplication and powers of powers follow.
For example, x^2 · x^3 = (x · x) · (x · x · x) = x^5 = x^(2+3)
(x^2)^3 = (x^2) · (x^2) · (x^2) . . . . by the basic idea of repeated factors
then ...
(x^2)^3 = x^(2+2+2) = x^(2·3) = x^6 . . . . by the rule for multiplication
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Division rules come in similar fashion.
(x^3)/(x^2) = (x · x · x)/(x · x) = x = x^(3 -2) . . . . denominator factors cancel a corresponding number of numerator factors.
Writing the reciprocal of this gives the rule for negative exponents:
(x^2)/(x^3) = (x · x)/(x · x · x) = 1/x = x^(2 -3) = x^-1 . . . . a power in the denominator is equivalent to a negative numerator power (and vice versa)
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Rules for fractional exponents derive from the multiplication rule.
(√x)(√x) = x^1 = (x^a)(x^a) = x^(2a) . . . . . . where √x = x^a
equating versions of the exponent, we see that a=1/2, so √x = x^(1/2).
Similarly, the n-th root is x^(1/n).
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The zero power rule comes from the division rule:
(x^2)/(x^2) = 1 = x^(2-2) = x^0 . . . . . for any non-zero x
