A zero mean random signal is uniformly distributed between limits –a and a and its mean square value is equal to its variance
This looks like homework, so here are some pointers :-

Firstly, you need to find the probability density function P(x) for the signal, given that :-

\$\int\limits_{-a}^{+a} P(x)\,\mathrm dx=1\$

The variance is given by :-

\$\int\limits_{-a}^{+a} x^2P(x)\,\mathrm dx\$

since the mean is zero.
The rms is the square-root of this by definition (as Olin says), so you have your answer in terms of a.