I'll occasionally write equalities within parentheses or sqrt signs to make my steps more compact.
E.g.:
$$ r = \frac{3}{4}\sqrt[4]{\frac{7}{9}\cdot\frac{16}{7}=\frac{16}{9}} = \frac{3}{4}\sqrt{\frac{4}{3}}=\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $$
I always assumed this was pretty clear, and an acceptable (if unusual) way to use equalities.
However, I was recently told that "you can't take the root of a truth value", which is certainly true.
Is my notation confusing? Does it classify as abuse of notation?
Is the below correct?
$$ r = \left(\frac{7}{9} \cdot \frac{16}{7} = \frac{16}{9}\right) \implies r = \top $$
Your notation is very non-standard and will be confusing to most people. Many people use curly braces to denote intermediate results: $$ r = \frac{3}{4}\underbrace{\sqrt[4]{\frac{7}{9}\cdot\frac{16}{7}}}_{\sqrt[4]{\frac{16}{9}}} = \frac{3}{4}\sqrt{\frac{4}{3}}=\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $$