To understand negative exponents one must understand the rules of fractions and operations of exponents.
First: any number or expression can be represented as a fraction with a denominator of 1.
A negative exponent is telling us to take the fraction to which it is being applied and rewrite it as its inverse ("flip it over"). The exponent then becomes positive. Any other operations signified by the exponent are then applied. Keep in mind, if the exponent is outside of a set of parenthesis, it is applied to what is inside; if it is applied to an expression without parenthesis it affects only the factor to its immediate left.
`x^-2 = 1/x^2`
We can apply these to the expressions from above to determine their correctness:
1. `10^-4 = 1/10^4 = 1/10*10*10*10 = 1/10000`
2. `a^2 b^-3 = a^2/1 * 1/b^3 = a^2/b^3`
3. `(5/3)^-3 = (3/5)^3 = 3^3/5^3 = (3*3*3)/(5*5*5) = 27/125`