Negative integer exponential power

The power of any non-zero number -n(n is a positive integer) is equal to the reciprocal of the power of this number, that is, a (-n) = 1/(a n).

Proof method

Prove: a (-n) = a (0-n) = a 0/a n, because a 0 = 1, a (-n) = a (0-n) =1a n, (a)

After introducing negative exponential power, the operational properties of positive integer exponential power (① ~ ⑤) are still applicable: (a m) (an n) = a (m+n) ①.

In other words, the powers of the same radix are multiplied, the radix remains the same, and the exponents are added.

(a^m)^n = a^(mn) ②

That is, the power of the power, the base is unchanged, multiplied by the index.

(ab)^n=(a^n)(b^n) ③

That is, the power of the product, multiplied by each factor.

(a^m)÷(a^n)=a^(m-n) ④

That is to say, the same base power is divided by the same base, and then the exponent is subtracted.

(a/b)^n=(a^n)/(b^n) ⑤

That is, fractional power, numerator and denominator are multiplied respectively.