Powers with Negative Integer Exponents
The power of a nonzero integer raised to a negative integer exponent is equal to the reciprocal of the base raised to the corresponding positive exponent. $$ b^{-n} = \frac{1}{b^n} $$
This rule can be understood by examining the division of two powers that share the same base.
For example, the division 23 : 25 can be written as a fraction
$$ 2^3:2^5 = \frac{2^3}{2^5} $$
Using the properties of exponents, the quotient of two powers with the same base is equal to the base raised to the difference of the exponents.
$$ \frac{2^3}{2^5} = 2^{3-5} = 2^{-2} $$
This follows from the definition of a power as repeated multiplication of the base by itself as many times as specified by the exponent.
$$ \frac{2^3}{2^5} = \frac{2 \cdot 2 \cdot 2 }{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2} $$
After simplifying the fraction into an equivalent fraction, we obtain the same result.
$$ \frac{2^3}{2^5} = \frac{\require{cancel} \cancel{2} \cdot \cancel{2} \cdot \cancel{2}}{\cancel{2} \cdot \cancel{2} \cdot \cancel{2} \cdot 2 \cdot 2} = \frac{1}{2 \cdot 2} = \frac{1}{2^2} = 2^{-2}$$
Therefore, raising a number to a negative integer exponent is equivalent to taking the reciprocal of that number raised to the corresponding positive exponent.
Powers with Negative Integer Exponents of Rational Numbers
The same principle also applies to rational numbers.
The power of a nonzero rational number raised to a negative integer exponent equals the reciprocal of that rational number raised to the corresponding positive exponent $$ (\frac{a}{d})^{-n} = \frac{1}{(\frac{a}{d})^n} = (\frac{d}{a})^n $$
This rule becomes clear through a simple example.
Consider the division (2/3)3 : (2/3)5 between two rational numbers with the same base, where the divisor has a larger exponent than the dividend.
$$ (\frac{2}{3})^3 : (\frac{2}{3})^5 $$
Rewrite the division as a fraction.
$$ (\frac{2}{3})^3 : (\frac{2}{3})^5 = \frac{(\frac{2}{3})^3}{(\frac{2}{3})^5} $$
According to the properties of exponents, dividing two powers with the same base yields a power with the same base and an exponent equal to the difference of the exponents.
$$ (\frac{2}{3})^3 : (\frac{2}{3})^5 = \frac{(\frac{2}{3})^3}{(\frac{2}{3})^5} = (\frac{2}{3})^{3-5} = (\frac{2}{3})^{-2} $$
This result can also be understood by recalling that a power represents repeated multiplication of the same base.
$$ (\frac{2}{3})^3 : (\frac{2}{3})^5 = \frac{(\frac{2}{3})^3}{(\frac{2}{3})^5} = \frac{ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} }{ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} } $$
We can now proceed with simplifying the fraction to obtain an equivalent fraction.
After simplification, the same result emerges.
$$ (\frac{2}{3})^3 : (\frac{2}{3})^5 = \frac{(\frac{2}{3})^3}{(\frac{2}{3})^5} = \frac{ \require{cancel} \cancel{ \frac{2}{3} } \cdot \cancel{ \frac{2}{3} } \cdot \cancel{ \frac{2}{3} } }{ \cancel{ \frac{2}{3} } \cdot \cancel{ \frac{2}{3} } \cdot \cancel{ \frac{2}{3} } \cdot \frac{2}{3} \cdot \frac{2}{3} } = \frac{1}{ \frac{2}{3} \cdot \frac{2}{3} } = \frac{1}{ (\frac{2}{3})^2 } $$
Finally, the division between the fractions 1/1 : (2/3)2 can be rewritten as the product of the first fraction and the reciprocal of the second fraction
$$ \frac{1}{ (\frac{2}{3})^2 } = \frac{1}{1} \cdot ( \frac{3}{2} )^2 = (\frac{3}{2})^2 $$
And the same reasoning applies in general.
