CRT-RSA is a variant of RSA, which uses integers d_p = d mod(p - 1) and d_q = d mod(q - 1) (CRT-exponents), where d,p,q are the secret keys of RSA. May proposed a method to obtain the secret key in polynomial time if a CRT-exponent is small, moreover Bleichenbacher and May improved this method. On the other hand, Takagi's RSA is a variant of CRT-RSA, whose public key N is of the form p^rq for a given positive integer r. In this paper, we extend the May's method and the Bleichenbacher-May's method to Takagi's RSA, and we show that we obtain p in polynomial time if $p < N^{3/(4 + 2 \\sqrt{r(r+3)})}$ by the extended May's method, and if $p < N^{6/(5r + \\sqrt{13r^2 + 48r})}$ by the extended Bleichenbacher-May's method, when d_q is arbitrary small. If r=1, these upper bounds conform to May's and Bleichenbacher-May's results respectively. Moreover, we also show that the upper bound of p^r increase with an increase in r. Since these attacks are heuristic algorithms, we provide several experiments which show that we can obtain the secret key in practice.