The laws of algebra, when applied randomly, will often make an expression more complicated to handle, rather than simpler to handle. This is because most of these laws are reversible: if a law allows you to reduce a complicated identity 𝐴=𝐵 into a simpler identity 𝐶=𝐷, it will also often allow you to expand the simpler identity 𝐶=𝐷 back into the complicated one 𝐴=𝐵.
So one has to apply the laws of algebra strategically. In high school one is taught some general principles for doing this, such as gathering like terms, eliminating variables, or factoring out (or cancelling) common factors, but in general strategy is not emphasized in most math courses.
A lot of mileage can be gained by performing local "gradient flow" on the complexity of the expressions at hand. For instance, all other things being equal, one would rather work with quotients with a complicated numerator and simple denominator than the other way around, because we have many more techniques to deal with messy numerators than messy denominators: for instance, (𝐴+𝐵)/𝐶 can be easily split into the sum of 𝐴/𝐶 and 𝐵/𝐶, but it is much less convenient to split 𝐴/(𝐵+𝐶) as the harmonic sum of 𝐴/𝐵 and 𝐴/𝐶, as we have so few laws for harmonic summation. This explains why it is often good to rationalize the denominator, but the trick is more general; for instance I recently was studying an expression involving the quantity 1/lcm(𝑛,𝑚) and I immediately knew to convert this to gcd(𝑛,𝑚)/𝑛𝑚 as there were many further manipulations I could make with the latter expression (in this case, the next step was to write ( \mathrm{gcd}(n,m) = \sum_{d|n,m} \phi(d)
) to begin decoupling the variables 𝑛 and 𝑚.)(1/3)
