The Laws of Exponents - Exploring Binary

For your reference, here is a summary of the laws of exponents:

$\mbox{\small{\displaystyle {x^0 = 1}}}$
$\mbox{\small{\displaystyle {x^1 = x}}}$
- Special case: $\mbox{\small{\displaystyle {1^{a} = 1}}}$
- Special case: $\mbox{\small{\displaystyle {x^{-1} = \frac{1}{x}}}}$
$\mbox{\small{\displaystyle {\frac{1}{x^{-a}} = x^{a}}}}}$
$\mbox{\small{\displaystyle {{x^a} \cdot {x^b} = x^{a+b}}}}$
$\mbox{\small{\displaystyle {\frac{x^a}{x^b} = x^{a-b}}}}$
$\mbox{\small{\displaystyle {\left({x^a}\right)^{b} = x^{ab}}}}$
$\mbox{\small{\displaystyle {{\left(x \cdot y\right)}^a = x^a \codt y^a}}}$
- Special case: $\mbox{\small{\displaystyle {\left({\frac{1}{y}}\right)^a = \frac{1}{y^a}}}}$
- Special case: $\mbox{\small{\displaystyle {x^{\frac{1}{b}} = \sqrt[b]{x}}}}$