In this paper, an adaptive approximation approach is proposed for the design of a divider and a square root (SQR) circuit. In this design, the division/SQR is computed by using a reduced-width divider/SQR circuit and a shifter by adaptively pruning some insignificant input bits. Specifically, for a $2n/n$ 2 n / n division, $2k$ 2 k and $k$ k ($k< n$ k < n ) consecutive bits are selected starting from the most significant ‘1’ in the dividend and divisor, respectively. At the same time, redundant least significant bits (LSBs) are truncated or if the number of remaining bits after pruning is smaller than the number of bits to be kept, ‘0's are appended to the LSBs of the inputs. To avoid overflow, a $2(k+1)/(k+1)$ 2 ( k + 1 ) / ( k + 1 ) divider is used to compute the $2k/k$ 2 k / k division. Finally, an error correction circuit is proposed to recover the error caused by the shifter using OR gates. For a $2n$ 2 n -bit approximate SQR circuit, similar pruning schemes are used to obtain a $2k$ 2 k -bit radicand. A $2k$ 2 k -bit SQR circuit and a shifter are then utilized to compute the SQR. This adaptive operation leads to very small maximum error distances of the approximate divider and SQR circuits, as shown by a theoretical error analysis. The proposed 16/8 approximate divider using an 8/4 exact array divider is $2.5\times$ 2 . 5 × as fast but only consumes 34.42 percent of the power of the accurate design. Compared to the accurate 16-bit array SQR circuit, the approximate design with a 6-bit radicand is $3.9\times$ 3 . 9 × as fast and consumes 20.66 percent of the power. The approximate SQR circuit using a 6-bit lookup table-based SQR circuit consumes 7.15 percent of the power of its corresponding accurate design. The proposed designs outperform other approximate designs in image processing applications including change detection (for the divider), envelope detection (for the SQR circuit) and image reconstruction (for both designs).