The paper starts with a short, intuitive revision of the derivation of the strictly positive constant \(\gamma\), the disconnection exponent for the two-dimensional Brownian motion. The right-hand side of the well-known double inequality \({1\over 2\pi} \leq \gamma \leq {1\over 2}\) is improved by the estimate \(\gamma \leq {1\over 2} - {1\over 2} \biggl( {\log 2\over \pi}\biggr)^2 1\). A well written derivation of \(\gamma_n\) precedes the proof of this second proposition. Using the similarities between disconnection and intersection exponents, some of the previously used arguments for disconnection exponents are modified to also provide some improved upper bounds for intersection exponents of planar Brownian motion. Several interesting conjectures regarding disconnection and intersection exponents are motivated throughout the paper.