In this paper, we consider the problem of finding how close two sums of mth roots can be to each other. For integers m≥2, k≥1 and 0≤s≤k, let em(s,k)>0 and Em(s,k)>0 be the largest exponents such that for infinitely many integers N there exist k positive integers a1,…,ak≤N for which two sums of their mth roots ∑j=1sajm and ∑j=s+1kajm are distinct but not further than N−em(s,k) from each other, or they are distinct modulo 1 but not further than N−Em(s,k) from each other modulo 1. Some upper bounds on em(s,k) and Em(s,k) can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that em(s,k)≥min(2s,k−1,2k−2s)−1/m for 1≤s<k and that Em(s,k)≥min(2s,k−2,2k−2s)+2−1/m for 0≤s≤k. Very recently, Steinerberger managed to show that E2(k,k)≥ck3, where c>0 is a small absolute constant. This seems to be the first result when the bound for s=k is increasing in k. By an entirely different argument, for any integers m≥2, k≥1 and s in the range 0≤s≤k, we show that Em(s,k)≥(k−2)/m+1. In particular, for m=2 and any non-negative integer s≤k, this yields the bound E2(s,k)≥k/2, which is much better than ck3. We also prove that e2(2,4)=7/2, which settles a problem raised by O'Rourke in 1981. These problems can be also considered for non-integer m. In particular, we show that 1≤E3/2(1,1)≤4/3, and that E3/2(1,1)=1 under assumption of the abc-conjecture.