For a nontrivial additive character \lambda and a multiplicative character \chi of the finite field with q elements (q a power of an odd prime), and for each positive integer r, the exponential sums \sum \lambda ((\tr w)^r) over w\in {SO}(2n+1,q) and \sum \chi (\det w)\lambda ((\tr w)^r) over {O}(2n+1,q) are considered. We show that both of them can be expressed as polynomials in q involving certain exponential sums. Also, from these expressions we derive the formulas for the number of elements w in {SO}(2n+1,q) and {O}(2n+1,q) with (\tr w)^r=\beta , for each \beta in the finite field with q elements.