Efficient initial approximations and fast converging algorithms are important to achieve the desired precision faster at lower hardware cost in multiplicative division and square root. In this paper, a new initial approximation method for division, an accelerated higher order converging division algorithm, and a new square root algorithm are proposed. They are all suitable for implementation on an arithmetic unit where one multiply-accumulate operation can be executed in one cycle. In the case of division, the combination of our initial approximation method and our converging algorithm enables a single iteration of the converging algorithm to produce double-precision quotients. Our new square root algorithm can form double-precision square roots faster using smaller look-up tables than the Newton-Raphson method.