### Theory:

Zero exponent
Any number raised to the power zero is equal to $$1$$. Generally, we write $$a^0 = 1$$, $$a \ne 0$$.
Example:
(i) $$2^0 = 1$$

(ii) $$100^0 = 1$$

(iii) $$15624^0 = 1$$

(iv) $$(-1)^0 = 1$$

(v) $$(-256)^0 = 1$$
Negative exponent
A number with negative exponent is equal to the reciprocal of the number with positive exponent.
That is, ${a}^{-m}=\frac{1}{{a}^{m}}$, where $$m$$ is an integer.
Example:
(i) ${\left(3\right)}^{-1}=\frac{1}{{3}^{1}}=\frac{1}{3}$

(ii) ${\left(5\right)}^{-3}=\frac{1}{{5}^{3}}$
If the negative number raised to the negative odd power, then the resultant value is negative.

If the negative number raised to the negative even power, then the resultant value is positive.
Example:
(i) ${\left(-5\right)}^{-3}=\frac{1}{{\left(-5\right)}^{3}}=-\frac{1}{125}$

(ii) ${\left(-4\right)}^{-5}=\frac{1}{{\left(-4\right)}^{5}}=-\frac{1}{1024}$

(iii) ${\left(-3\right)}^{-2}=\frac{1}{{\left(-3\right)}^{2}}=\frac{1}{9}$

(iv) ${\left(-2\right)}^{-4}=\frac{1}{{\left(-2\right)}^{-4}}=\frac{1}{16}$
Important!
$$(-1)^{\text{odd number}} = -1$$

$$(-1)^{\text{even number}} = 1$$